July 26th, 2012, 11:45 AM  #1 
Newbie Joined: Jun 2012 Posts: 13 Thanks: 0  Sum of Odd Integers
Let be odd integers. Prove that , where the sum is from i=1 to i=k, is odd iff k is odd. Thanks for help! 
July 26th, 2012, 11:49 AM  #2 
Global Moderator Joined: May 2007 Posts: 6,583 Thanks: 610  Re: Sum of Odd Integers
Easy by induction: Even number + odd number is odd, odd number + odd number is even. k=1, sum is odd, k=2, sum is even, and odd  even alternate as you add each term. 
July 26th, 2012, 12:00 PM  #3 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: Sum of Odd Integers
If k is even, then you can split the numbers into pairs, and each pair of odd numbers adds to an even number. Since all of these numbers are divisible by 2, the sum of all the numbers is also divisible by 2, and is therefore even. If k is odd, then you can split the numbers into pairs as before, except you will have one left over. The sum of all the numbers put into pairs is even, and then adding the leftover number, which is odd, gives you an odd total. 
July 27th, 2012, 04:34 AM  #4  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Sum of Odd Integers Quote:
So, the sum S is odd if and only if k is odd. Every term of can be represented as So, rewrite the sum using the above representation : so, is odd if and only if S is odd and S is odd if and only if k is odd implies is odd iff k is odd  
July 27th, 2012, 01:28 PM  #5 
Newbie Joined: Jun 2012 Posts: 13 Thanks: 0  Re: Sum of Odd Integers
Thanks for the responses, very helpful.

July 27th, 2012, 02:29 PM  #6 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Sum of Odd Integers
The way I would is to note that we can write any odd number in the form "2k+1" for some integer k. That means we can write the so that the sum is . The first sum is even because of the "2". If n is even, this is the sum of two even numbers so even. If n is odd, this is the sum of an even and an odd number so is odd.

July 29th, 2012, 07:05 AM  #7  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Sum of Odd Integers Quote:
 
July 29th, 2012, 07:38 AM  #8 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Sum of Odd Integers
In some sense all methods are equivalent. But it's a somewhat different expression. The hope, presumably, is that the OP would find one of these expressions more understandable than the others.

July 31st, 2012, 12:09 AM  #9 
Newbie Joined: Jul 2012 Posts: 2 Thanks: 0  Re: Sum of Odd Integers
It is difficult for me to understand it.

July 31st, 2012, 03:39 AM  #10  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Sum of Odd Integers Quote:
 

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