Formula for Triangular NumbersDate: 08/29/98 at 17:32:48 From: Tobin Huff Subject: Formula for triangular numbers Is there a formula to figure out triangular numbers? With square numbers, the formula is the number of dots on one side squared: ... ... ... is 3^2 = 9. If my question isn't clear enough, please write back and I will try to reword it. Thank you, Tobin Huff Date: 08/29/98 at 23:23:44 From: Doctor Barrus Subject: Re: Formula for triangular numbers Hi, Tobin! There is indeed a formula for triangular numbers. The Nth triangular number is calculated by: N(N + 1) -------- 2 We find this by examining the patterns in triangular numbers. Here are the first few triangular numbers: * 1 = 1 * * * 3 = 1 + 2 * * * * * * 6 = 1 + 2 + 3 * * * * * * * * * * 10 = 1 + 2 + 3 + 4 You can see that to get the Nth triangular number, we just add all the positive integers through N. Now let's find a formula for this sum. Look what happens if we write the first N integers in ascending order in one row, and then in descending order in the next row, and then add the numbers in each column: 1 2 3 ... N N N-1 N-2 ... 1 --- --- --- --- N+1 N+1 N+1 N+1 Each column adds up to N+1. Now add these sums horizontally. Since there are N of them, and they're all equal to N+1, we get N(N+1). Notice what we've done here. We've added all the numbers from 1 to N twice: (1 + N) + (2 + N-1) + (3 + N-2) + ... + (N + 1) = N(N+1) = (1 + 2 + 3 + ... + N) + (N + N-1 + N-2 + ... + 1) So to get the sum of the first N positive integers, we'll just divide by two. Then: 1 + 2 + 3 + ... + N = N(N+1)/2 And this is our formula for triangular numbers. I hope this has made sense. Good luck! - Doctor Barrus, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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