Formula for Triangular Numbers

Date: 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/