Associated Topics || Dr. Math Home || Search Dr. Math

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/
```
Associated Topics:
High School Sequences, Series

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search