Fascinating numbers

They say one man's garbage is another man's treasure. Well what may be one man's tedium is my fascination.

My latest infatuation with numbers stems from the simple fact that:

1^2 =   1 =  1
2^2 =   4 =  1 + 3
3^2 =   9 =  1 + 3 + 5
4^2 =  16 =  1 + 3 + 5 + 7  
5^2 =  25 =  1 + 3 + 5 + 7 + 9

I know it doesn't seem like much to be excited about but I can't stop thinking about this. It's as if there lies a secret in this information, a secret that I have to find. I haven't found it yet but I will keep looking!

By the way, don't you think that's just the coolest thing? n^2 = the sum of the first n odd integers

Debbie's fascination that the sum of the first n odd numbers is n² is understandable, but this is also easy to prove. The key is that the "triangle sum", the sum of the numbers from 1 to n, is equal to n(n+1)/2. (See note at bottom for a quick visual proof.) The triangle sum is easy to demonstrate by induction; it is often used in a classroom setting as an introductory example of how induction works. In fact it's proved in a writeup in the induction node. Hey wait a minute, it's proved in Debbie's writeup in the induction node.

So Debbie--you already know a way to demonstrate this.

n2 = 1 + 3 + 5 + 7 + ... + 2n-1

   = 1+2+3+4+5+6+7+ ... + 2n-1       (sum of all integers up to 2n-1)
   -   2 + 4 + 6 + ... +2n           (less the even integers up to 2n)

But we can represent both of those as a triangle sum T: the first is just T_2n-1, the sum of the numbers from 1 to (2n-1), and the second, the sum of the even integers up to 2n is just 2T_n, twice the sum of the integers up to n:

   = T2n-1 - 2*Tn-1
  
     (2n-1)*2n       (n-1)*n
   = ---------  - 2*(--------)
         2              2
  
   = (2n-1)*n  - (n-1)*n
  
   = (2n-1-n+1)*n
  
   = n*n = n2

Pretty neat, but now you need something else to ponder. How about the sum of the first n integers squared (1 + 4 + 9 + ...)? That's trickier, but doable. See the first couple chapters of Concrete Mathematics by Knuth et al if you get stuck; they do a great job with it.

(NOTE: T_n is called the "triangle sum" because it can be visualized as a triangle. Imagine a set of bowling pins arranged as:

      o
      o o
      o o o
      o o o o
      o o o o o
      ...

The number of pins is clearly 1 + 2 + ... + n, where n is the last row. Now put that triangle together with another triangle arranged upside-down, and you get an n x (n+1) rectangular matrix of pins. So the number of pins in the triangle is half that, n*(n+1)/2.)