First 500 Even/Odd Numbers

Date: 03/14/2002 at 15:11:58
From: Roxanne Bossert
Subject: Math

I would like to know if there is a formula that will give me the 
answer to the difference of the sum of the first 500 even numbers and 
the sum of the first 500 odd numbers.

Thanks.

Date: 03/14/2002 at 17:32:09
From: Doctor Ian
Subject: Re: Math

Hi Roxanne,

If I understand you correctly, you want to find

  ? = (2 + 4 + 6 + ... + 1000) - (1 + 3 + 5 + ... + 999) 


Note that we can play with this a little:

  ? = 2 + 4 + 6 + ... + 1000 - 1 - 3 - 5 - ... - 999

    = 2 - 1 + 4 - 3 + 6 - 5 + ... + 1000 - 999

    = (2 - 1) + (4 - 3) + (6 - 5) + ... (1000 - 999)

Does this help? 

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

Date: 03/20/2002 at 18:46:37
From: Michelle
Subject: Problem Solving

I was hoping you would be able to help me figure this problem out by 
showing me the most simple way to figure it out and by showing me 
every step.

Odd and Even: Find the difference between the sum of the first 500 
even numbers and the sum of the first 500 odd numbers.

Thanks in advance! 
Michelle

Date: 03/20/2002 at 21:14:40
From: Doctor Jubal
Subject: Re: Problem Solving

Hi Michelle,

Thanks for writing Dr. Math.

The first odd number is 1.  The first even number is 2.  Their 
difference is

  (2-1) = 1

Now, what about the difference between the sums of the first two even 
and odd numbers?

  (2+4) - (1+3) = 6 - 4 = 2

But it might be a little more enlightening if we wrote it this way...

  (2-1) + (4-3) = 1 + 1 = 2

So the difference between the sum of the first 500 evens and the first 
500 odds is

  (2-1) + (4-3) + (6-5) + (8-7) + ... + (1000-999) = ?

Do you see the pattern?  Write back if you'd like to talk about this 
some more, or if you have any other questions.

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