Sums of Consecutive Odd vs. Even Integers

Date: 04/15/2002 at 18:51:42
From: Charles
Subject: Math Portfolio

Can the sum of two consecutive even integers ever equal the sum of 
two consecutive odd integers? Why or Why not?

Date: 04/15/2002 at 23:29:45
From: Doctor Ian
Subject: Re: Math Portfolio

Hi Charles,

Any odd integer can be written as 

  2*k + 1

where k is some integer. Try some examples, to convince yourself that 
it's true. (For example, 7 is odd; and it can be written as 2*3 + 1.)

That means that we can write any two consecutive odd integers as 

  2*k + 1 and 2*k + 3

If we add these together, we get 

  2*k + 1 + 2*k + 3 = 4*k + 4

Following the same kind of reasoning, what happens if you add two 
consecutive even integers together? Can you ever get something that 
looks like 4*k + 4?

I hope this helps. Write back if you'd like to talk more about this, 
or anything else.

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