Hello, hello children! The best chapter of the world of Maths, Sequences and Series, we are learning this. I hope your concepts are becoming clear in this. And I hope your studies are going well. If you keep working hard, God will make you successful. If you keep working hard, God will make you successful. And if you don't study, life will become "Zingala La La". If you keep working hard, God will make you successful. And if you don't study, life will become "Zingala La La".

In this session, we are going to see Geometric Progression, okay? GP. We will see these two things. We will see Geometric Progression, GP, and then we will see infinite GP. If you have learned all the concepts of GP and all the concepts of infinite GP, then understand that you have understood things clearly in this lecture. If you don't calculate the sum of infinite GP, then life is "Zingala La La". Then what is the point?

So, how are you all? I hope you all are well, and I am also absolutely fine. So, let's start the lecture. Geometric Progression is our topic. GP, GP is our topic. It is also called Geometric Progression. GP. Geometric Progression. In the previous lecture, we saw AP. What did we see? AP. You said Alakh Pandey? No, Arithmetic Progression. 2012 Arithmetic Progression.

We saw Arithmetic Progression. Okay. What is in GP? In GP, there is a sequence, okay? There will be a first term, listen. There will be a first term, 'a'. And if we multiply it with 'r', it becomes 'ar'. See, if we multiply with 'r', it becomes 'ar'. If we multiply 'ar' with 'r', it becomes 'ar square'. If we multiply 'ar square' with 'r', it becomes 'ar cube'. By multiplying 'r' again and again, by multiplying 'r' again and again, this sequence that develops is called GP. What is it called? GP. This sequence that develops is called GP. And the first term is 'a'. Understand this. The first term is 'a'. Now look carefully. The first term is 'a'. Look carefully, this is a very important topic. The first term is 'a'.

And when we multiply with 'r', we get the next term. And by multiplying like this with this number, this special number with which we keep multiplying, this special number, with which we keep multiplying, this number is called the Common Ratio. What is it called? Common Ratio. So, if you know two things in GP, if you know the first term and the common ratio, then you can know the entire GP. You can know every bit about GP if you know two things: the first term and the common ratio. What is the first term? 'a'. What is the common ratio? 'r'. What does common ratio mean? Why is it called common ratio? Let's see.

If you look carefully, what is term number one? Term number one is 'a'. What is term number two? 'ar'. What is term number three? 'ar square'. Term number four? 'ar cube'. Term number five? 'ar power four'. Term number six? 'ar power five'. See the interesting thing. If I divide T2 by T1, if I divide T2 by T1, then I will get 'r'. If I divide T3 by T2, then 'ar square' divided by 'ar', then again I will get 'r'. If I divide T4 by T3, then again I will get 'r'. See here, what is the point to understand? If you divide T2 by T1, T3 by T2, T4 by T3, the ratio always remains fixed, and it is equal to 'r'.

So, this first point we have understood. This point we have understood. If we divide T2 by T1, if we divide T3 by T2, if we divide T4 by T3, then 'r' will remain the same. It will be 'r'. Meaning, the ratio of two consecutive terms is always constant and equal to 'r'. Understood? So, T2 by T1, T3 by T2, T4 by T3. This 'r' remains equal. It is always equal to 'r'. And see this 'r', my dear, this is that 'r'. What is it called? What is it called? Common Ratio. See, my dear, this is that 'r'. What is it called? Common Ratio. So, in GP, if you know the first term and the common ratio, then you can create a whole storm in GP. Okay?

This story is clear. Now, the second thing is, if I say, tell me the 10th term of GP. Which term? The term that comes at the tenth number. What will you say? The first is 'a', the second is 'ar', the third is 'ar square', the fourth is 'ar cube', the fifth is 'ar power four', the sixth is 'ar power five', the seventh is 'ar power six', the eighth is 'ar power seven', the ninth is 'ar power eight', the tenth is 'ar power nine'. See, isn't it coming? Observation. Observation. So, what will the 20th term be? 'ar power 19'. The nth term will be 'ar power n-1'. If it is the 10th term, then it will be 'ar power 9'. If I ask for the 555th term, it will be 'ar power 554'. And if I ask for the 10th term, you can say it will be 'ar power n-1'. This is 'n', so here 'n-1' will come. Because this is 10, so this is 9. So, you can see the question there. So, the nth term, you can call it the general term. So, we first learned to find the general term. If you need to find the nth term of GP, you can find it using the formula 'ar power n-1'. Okay? If you need to find the nth term, the nth term of GP, then it will be 'ar power n-1'. Okay? You can call it the nth term or the general term. What else should I say? General term. Okay.

So, you got the idea of GP. The ratio of two consecutive terms is the same, my dear, the common ratio. And the nth term can be found using the formula 'ar power n-1'. Okay? You can find it using 'ar power n-1'. Okay? Now, see, what I am saying is that it is a sequence. So, tell me, is this a GP? Yes, absolutely, it is a GP. Because if you multiply it by 3, then multiply it by 3, then multiply it by 3, and by doing this, you will get the next term. And the terms of this GP are increasing. So, this is an increasing GP. Okay? Now, look at this one. 15, 15/3, 15/9, 15/27. Like this, by multiplying by 1/3, you get the next terms. See. So, its 'r' is 1/3. If you multiply this by 1/3, you get this. If you multiply this by 1/3, you get this. This continues. So, 'r' is 1/3. And see this GP. The terms are decreasing here. They are decreasing. So, this will be called a decreasing GP. What will it be called? It will be called a decreasing GP. This is our decreasing GP. A noteworthy thing is that for an increasing GP, its 'r' is greater than 1. If you keep multiplying by a quantity greater than 1, things will increase. Understood? And here, 'r' is less than 1, between 0 and 1. Okay? Let's take an example. Tell me, I am writing it. Tell me if this is a GP. You can call it a constant GP. This is a GP. It is a constant GP. What is its 'r'? Its 'r' is 1. Keep multiplying by 1. Keep multiplying by 1. You can call this an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. What is its 'r'? Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. 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You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. You can call it a constant GP. Its 'r' is 1. Keep multiplying by 1. You can call it an AP. You can call it an AP. You can call this sequence an AP. You can call this sequence a GP. 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Salman, if there is a GP, then CR squared, a cube, love to the power of four, friend to the power of five, minus how many, I explained it well in the corrective rap, didn't I? Take 34 numbers from the GP, take four, take five matches, then you can approve it like this, okay. Fight. Now, friend, let's talk about the north point, and very importantly, how to find it. Now let's talk about how to find the sum of GPs, and here I will tell you, we will talk about the general sum, we will talk about the sum of GPs, okay. How to find the sum of n terms, how to find the sum of n terms, how to find the sum of n terms of GP. Yes, how to find the sum of n terms of GP. We have a GP, it is A and CR, friend, squared, execute, and so on, up to the power of minus two, and the nth term will be A to the power of R. To see this GP, we have to find the sum of its n terms. We have to find the sum of its n terms. Let's call it S.

Okay, there is a very interesting way to find the sum of a GP, a special way to find the sum of a GP. See what that way is. What I did is, I am interested in its value. Okay. I multiplied both sides by R. Multiply both sides by R. When I multiply both sides by R, how much is this? RS. Right side, if I multiply by R, how much is it? A R, A R squared, A R cubed, A R to the power of four, A R to the power of n minus one, A R to the power of n. Multiply both sides by R, so RS equals this quantity. I ask you one thing, if I write this quantity, that is, this constant, slightly shifted, will it make any difference? I don't think it will make any difference. Whether you write it here or write it there, it's the same thing. What's the problem? I wrote it shifted, but just by writing it shifted, the exam will be over. See, just by writing it slightly shifted, something amazing will happen. What amazing thing will happen? I'll tell you why. What I did is, I assumed the sum of the terms of the GP to be equal to this, and multiplied both sides by R. Okay. I got an equation, I got an equation, I subtracted equation two from one, meaning I subtracted this equation from this one. So what happens? When we subtract, the signs change. So the left side becomes S, and if we take it common, 1 minus R will remain. Here, R and minus AR cancel out. AR squared minus AR squared, AR cubed minus AR, all these cancellations happened. You see, all these cancellations happened. Only one will remain here. And 1 minus R to the power of n will remain. Minus A R to the power of n will remain. You see, it has arrived, my friends, very quickly. If I take A common from here, it will be 1 minus R to the power of n, and this will be A into 1 minus R to the power of n upon 1 minus R. Very wonderful, very wonderful, the formula for the sum of n terms of GP has arrived. You see, the formula for the sum of n terms of GP is A into 1 minus R to the power of n upon 1 minus R. You can also write it as A into R to the power of n minus 1 upon R minus 1. If you multiply by minus one above and below, you get R minus 1. So you have to put it here. So this formula has arrived. Quickly remember this formula. If you don't remember it, it will be a difficult day. Okay. Remember this formula, friend. What is R? It is the sum of n terms, the sum of n terms of GP. What is A in this? A is the first term. R is the common ratio. N is the number of terms. The meaning of these things should be clear. A is the first term. A is the first term. R is the common ratio. The common ratio. N is the number of terms. Did you understand? And the sum will be A into 1 minus R to the power of n upon 1 minus R. Let's use this formula and find the sum of a GP. Let's use this formula and find the sum of a GP. Look.

1 plus 2 plus 2 squared plus 2 cubed plus 2 to the power of 102. Okay. If I say, is this a GP? You will say, yes, it's a GP, no problem. If I ask for the sum, you will say, apply the formula. The formula for the sum is R to the power of n minus 1 upon R minus 1. What is the formula for the sum? I'll write it here. The formula for the sum is A R to the power of n minus 1 upon R minus 1. Okay. Let's apply the formula for the sum. What is A? It is the first term, 1. What is R? The common ratio, by multiplying by what are we moving forward? R is 2. What is the number of terms? Don't get confused, like you get confused in R. The number of terms here is not 100. It is 101. You have to understand this. The number of terms is 101. Oh, my dear, from 1, 2, 3, 4, 5, up to 100, these are the terms, and this one is also there. So there are 101 terms. So let's put it in. What is A? 1. What is R? 2. What is N? 101. Minus 1 upon R? What is it? 2. Minus 1. So this is the sum. This is the sum. 2 to the power of 101 minus 1. The answer is 2 to the power of 101 minus 1. 2 minus 1 is 1. So it is easy to find the sum of a GP, my friends. Okay. Let's move on. Let's look at another question. When I said, find the sum of the sequence, I said, find the sum of the series. I have a series. I have a series. And I said, find the sum of this series. Find the sum of this series. Up to when? Find the sum of 100 terms. Up to when? Find the sum of 100 terms. I have a series. Is it a GP? It's not a GP. Why is it given? There must be a reason. The friend gave it. It's not a GP. How did you know? Multiplying it by 11 gives 99. But multiplying it by 11 does not give 99. It's not a GP. It's not a GP. But it can be converted into a GP. This is an important point. It is not a GP. It is not a GP. But it can be converted into a GP. My friend, your traffic jam. Let me see. 9 plus 9 plus 9 plus 4. We are talking about 90 terms. Look here. The first term, the number nine, we can write it as 10 minus 1. Let's write 9. Let's write 10 squared minus 1. Let's write 9. Let's write 10 cubed, meaning 1000 minus 1. Let's write 10. Let's write 10 to the power of 4 minus 1. The series will continue. So the 100th term will be 10 to the power of 100 minus 1. Will it be 10 to the power of 100 minus 1? Do you understand, friend? Do you understand, friend? Let me tell you again. Let me tell you again. This is written as 10 minus 1. This is written as 10 squared minus 1. This is written as 10 cubed minus 1. This is written as 10 to the power of 4 minus 1. And this continues. What I did is, I need to find its sum. What I did is, I made two brackets. I said, isn't it 10 minus 1? Let's put 10 here and minus 1 there. 12 minus 1. Let's put 10 squared here. Minus 1 there. So here it became 8 plus 1. We have to write it. Minus 1. Let's collect 10 here. Minus 1. The series continued up to 10 to the power of 100. Here it became this collection, and here it became a collection of 1111. How many times did this 1 come? As many brackets as there were, that many ones came. There were 100 brackets. 101 came. 90 came. It is written as 100 terms. Okay. You can see, this was not a GP. It was broken and molded. It was broken and molded. Look, it has become a GP. It has become a GP. And one more thing here, this is also like a GP. Because its sum can be found. What is the common ratio? It is increasing by multiplying by 10, 10. The first term is this. It is 10. R is 10. N is 100 terms. So, yes, we can find it. This is A into R to the power of n minus 1 upon R minus 1. And minus. When you sum this bracket, when you wrap it up, it will become 100. How much will it become? 100. If you do 1 plus 1 plus 1, 100 times, it will become 100. So, yes, the sum of these terms minus 1. So 10 by 9 into 10 to the power of 100 minus 1 minus 100. The answer has arrived. So, you will simplify this story. You will simplify this story. After simplifying, you will get this. Did you understand? The answer is 10 by 9 into 10 to the power of 100 minus 52 minus 100. Some are directly given as GPs. Some may not be directly given as GPs. Sequences like this are given. I'll give you a homework question. First of all, if you have understood everything so far, tell me in the comments that you have understood everything. My heart will be at peace. And if you have understood everything, try this homework question. Many people tried the homework I gave in the previous session. I will discuss it. Many people have the correct answer. Salute to them. Okay. So this is your question, homework.

Find the sum of n terms of the series. Find the sum of n terms of the series. 1 plus 6 plus 77 plus 66. Find the sum of n terms of this series. Okay. If you can do it, definitely tell me in the comments that you have done it. And if you can, send the link to the solution. That would be great. It would be fun. Okay. This is your homework. So definitely comment. Even if you can't do it, comment. Definitely tell me. You definitely comment. Even if you don't comment, it's important. When you comment that I couldn't do it, it's important to know because then I know that you are not doing it, you are trying. But trying is important. Okay. So definitely try this question. Okay. I hope you understood the game of GP. Okay. In this session, we will cover the theory part completely. Now I will tell you about infinite GPs. This is a very cool thing. If you don't find the sum of infinite GPs, life will be meaningless. Somewhere, some great poet has said, who is that great poet? You know. If you don't find the sum of infinite GPs, life will be meaningless. What is an infinite GP? I will give you two infinite GPs. Remember. I am writing them in front of you. Look at these two infinite GPs. This is a GP, and it will continue, friend, it will never stop. And this is also a GP. This is also a GP, my friends. This is also an infinite GP. This is also an infinite GP. If you look at its sum, its sum will be infinity. A very large number. Why will it be? Because 2 is added to 1, then 4 is added, then 8 is added. Larger numbers will be added. Larger numbers will be added. And in the end, it will become a very huge number, a very large number, it will become infinity. But the answer to this sequence will not be infinity. The answer to this sequence will not be infinity. Because in the first term, 1/2 is added. Then a small 1/4 is added. Then even smaller is added. Then look, 1/8 is added. That is, small things are being added. Here, small things are being added. The answer will be finite. And I will tell you, its answer is 2. How it is 2, I will show you. But first, know this. This was an increasing GP. This is a decreasing GP. This was an increasing GP. This is a decreasing GP. The sum of an increasing GP is infinity. The sum of a decreasing GP is finite. The problem is, this is an interesting GP. Terms are added, large terms are added continuously. Okay, I am talking about positive terms. Okay. And look, this is a decreasing GP. So the sum of an increasing GP is infinite. What do we need to study? In GPs, I will tell you that when R is greater than 1. Okay. I am talking about the case when R is greater than 1. Here R is 1/2, which is less than 1. Here, its R is 2. R greater than 1. The answer will not be finite. R will be less than 1. If R is less than 1, between minus 1 and 1. The sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1. The answer will not be finite. If R is less than 1, between minus 1 and 1. If R is between minus 1 and 1, the sum of that GP, that series, will be finite. So, first of all, let's write this. The sum of an infinite GP will be finite. The sum of an infinite GP will be finite if R is greater than minus 1 and R is between minus 1 and 1. Okay. Here, look, what is R? It is 1/2. It is less than 1. Here, what is it? Its R is 2. R greater than 1.

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Sequence & Series 03 | Sum of Infinite G.P. | CLASS 11 | JEE | PACE SERIES

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