Summing Triangle NumbersDate: 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/ |
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