Sum of Odd Numbers 1-50?

Date: 10/10/2001 at 01:09:51
From: Shaundra
Subject: Problem Solving Looking for patterns

What is the sum of the first 50 odd numbers?

Date: 10/10/2001 at 13:40:44
From: Doctor Ian
Subject: Re: Problem Solving Looking for patterns

Hi Shaundra,

The sum of the first 5 odd numbers is

  1 + 3 + 5 + 7 + 9 = [sum]

Twice this sum is

  (1 + 3 + 5 + 7 + 9) + (1 + 3 + 5 + 7 + 9) = [sum] * 2

We can rewrite it this way:

  (1 + 9) + (3 + 7) + (5 + 5) + (7 + 3) + (9 + 1) = [sum] * 2

     10   +    10   +    10   +    10   +    10   = [sum] * 2

                                           5 * 10 = [sum] * 2

                                     (5 * 10) / 2 = [sum]

                                                  = 25

All I did was match the numbers up in reverse order:

    _____________________________________
   |    _____________________________    |
   |   |    _____________________    |   |
   |   |   |    _____________    |   |   |
   |   |   |   |    _____    |   |   |   |
   |   |   |   |   |     |   |   |   |   |
  (1 + 3 + 5 + 7 + 9) + (1 + 3 + 5 + 7 + 9) = [sum] * 2

This is a nice trick, because each pair adds up to the same amount -- 
the sum of the largest and smallest numbers in the list!

Can you see why it's as quick to do this for 50 numbers - or 500, or 
5000 - as it is for 5? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/