Transcription
Let's add up the first few cubes. 1 cubed is 1. 1 cubed plus 2 cubed, that's 9. 1 cubed plus 2 cubed plus 3 cubed, that's 36.
Now, add up the first few integers themselves and square the result. 1 squared is 1. 1 plus 2 squared, that's 9. 1 plus 2 plus 3 all squared, that's 36.
They're the same every time. The sum of cubes is the square of the sum. Now, I can't pronounce the name of this theorem, but here's the name of the theorem, and the proof is pretty cool and visual.
Imagine laying out 1 plus 2 plus 3 all the way up to n dots in a staircase pattern to make a triangle. That triangle has n * n + 1 over 2 dots. Square that triangle, literally making it into a square. The area of that square is the quantity n * n + 1 over 2 all squared.
And you can partition that square into L-shaped bands, where each band has exactly area k cubed. So, the cubes aren't just related to the sum, they are the sum squared.