Addition Shortcut

Date: 05/27/97 at 11:20:08
From: Anonymous
Subject: Math problem

Dear Dr. Math,

I have a problem that I know how to solve, but I would like to know if 
there is a shortcut.  The problem is:

	(2+4+6+8+ ... +34)/(3+6+9+12+ ... +54)

I solved it by adding all numbers up.  Please teach me a proper way of 
solving this kind of question.

Thank you very much,
Irene

Date: 05/27/97 at 18:57:31
From: Doctor Wilkinson
Subject: Re: Math problem

Dear Irene,

Adding all the numbers up is not stupid.  Lot's of people are too lazy 
to do that much.  But I'll show you a clever trick.  You want to add:

 2 + 4 + 6 + 8 + ... + 34

Do it twice!  This gives you: 

 2 + 4 + 6 + 8 + ... + 34   +

 2 + 4 + 6 + 8 + ... + 34

Now reverse the bottom row.  This gives you:

   2 + 4 + 6 + 8 + ... + 34  +
  34 +32 +30 +28 +...  +  2

= 36 +36 +36 +36 +...  + 36

You have seventeen 36's to add to get twice the sum you want, so the 
sum is:

 17 * 36 / 2 = 17 * 18

I'll let you figure out the sum in the denominator.

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/