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### Sum of Odd Numbers 1-50?

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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?
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```
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/
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