Transcription
Take a look at this problem. You have an infinite sequence of square roots nested inside of each other forever. It looks like a nightmare. If you saw it on a test, you'd probably skip it.
But, I'm going to show you how to solve this in under 60 seconds, and you don't even need a calculator. First, set the entire thing equal to X. So, X equals the square root of 2 plus the square root of 2 and so on repeating forever.
Now, here is where I want you to look really closely. I'm going to ask you a question. What exactly is this inner piece right here? Hmm. If the sequence literally goes on to infinity, then ignoring the very first layer does not actually change anything. That infinitely nested inner piece is the exact same thing as our original X. So, let's just replace it.
Suddenly, the entire infinite nightmare completely collapses into a simple equation. X equals the square root of 2 plus X. Now, we just do some basic algebra. Let's square both sides to get rid of that annoying square root symbol, and now we get X squared equals 2 plus X. Let's move everything over to the left side to create a standard quadratic equation. We subtract X, and we subtract 2. And now we have X squared minus X minus 2 equals 0.
That is super easy to factor. We're looking for two numbers that multiply to -2 and add to -1. That is -2 and positive 1. Wait, let me check. Yeah, that's correct. So, we get X minus 2 * X + 1 = 0. This gives us two possible answers. X could be 2 or X could be -1.
But, look back at our original problem. It's a sequence of positive square roots. A positive square root can never equal a negative number. So, we throw away -1 directly into the trash, and just like that, we have our final answer. X = 2. Our infinite nightmare is just 2.