Sequence of Triangular Numbers

Date: 13 Jul 1995 09:46:53 -0400
From: richard rudolph
Subject:Sequence

What is this sequence called:
  
1,3,6,10,15 

How is it generated?

Richard Rudolph   :-)

Date: 13 Jul 1995 22:39:00 -0400
From: Dr. Ken
Subject: Re: Sequence

Hello there!

This sequence is called the sequence of Triangular Numbers.  It's called
that for the following reason:

 *    *      *        *          *            *
     * *    * *      * *        * *          * * 
           * * *    * * *      * * *        * * * 
                   * * * *    * * * *      * * * * 
                             * * * * *    * * * * * 
                                         * * * * * *
                                                        . . . .

(count the stars)

Notice that if you count the number of stars in a triangle by counting the
number of stars in each row and then adding them, you'll get something like
this (this is for the last triangle): 1+2+3+4+5+6 = 21.  In general, if you
add up the first n positive integers, you'll get the number n(n+1)/2.  Do
you see why?  If you want to know why, you can write back to us (but try to
figure it out for yourself first, it's pretty neat!).

Thanks for the question!

-K