Summing Triangle Numbers

Date: 04/21/98 at 15:05:43
From: Karen Hooton
Subject: Sum of triangle numbers

Dear Dr. Math,

I am trying to find the formula for the sum of triangle numbers, such 
as: 1 + 3 + 6 + 10 + 15. 

I've gone through my school library and checked out the Internet with 
no luck. Is it at all possible that you can help me out?

Yours sincerely,
Karen Hooton

Date: 04/22/98 at 10:51:30
From: Doctor Nick
Subject: Re: Sum of triangle numbers

Hi Karen -

This is a nice question and it has a nice answer.

Let g(n) be the n-th triangular number. So:

     g(1) = 1
     g(2) = 3
     g(3) = 6

and so on. You probably know that:

     g(n) = n(n+1)/2

Let f(n) be the sum of the triangular numbers 1 through n.
So:

     f(n) = g(1) + g(2) + ... + g(n)

Then:
 
     f(n) = n(n+1)(n+2)/6

How can we prove this? We can prove it by induction. That is, we'll 
prove two things:
 
   1) It's true for some n (n=1, in this case).
   2) If it's true for n, then it's true for n+1.

This will allow us to conclude that it's true for all n >= 1.

Now 1) is easy. We know that f(1) = g(1) = 1. So it's true for n = 1.

Now for 2). Suppose it's true for n. Consider f(n+1). We have:

     f(n+1) = g(1) + g(2) + ... + g(n) + g(n+1) 
            = f(n) + g(n+1)

Using our assumption that f(n) = n(n+1)(n+2)/6 and that 
g(n+1) = (n+1)(n+2)/2, we have:

     f(n+1) = n(n+1)(n+2)/6 + (n+1)(n+2)/2
            = n(n+1)(n+2)/6 + 3(n+1)(n+2)/6
            = (n+1)(n+2)(n+3)/6

which is exactly what the formula says it should be. Thus we have 
shown that if it's true for n, it's true for n + 1. Since we showed it 
was true for n = 1, we now know it's also true for n = 1 + 1 = 2, and 
then for n = 2 + 1 = 3, and so on, for all n >= 1.

By the way, these numbers (1,4,10,20,35,56,...) are known as 
tetrahedral numbers. If you imagine piling triangles of dots on top of 
one another first 1, then 3, then 6, then 10 you'll get a tetrahedron 
of dots at each stage with 1,4,10,20 dots total. Take a look at:
  
  http://www.astro.virginia.edu/~eww6n/math/TetrahedralNumber.html   

for some high level information about them and related fun things.

Have fun,

-Doctor Nick,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/