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July 26th, 2012, 12:45 PM   #1
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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.

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July 26th, 2012, 12:49 PM   #2
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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.
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July 26th, 2012, 01:00 PM   #3
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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.
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July 27th, 2012, 05:34 AM   #4
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Re: Sum of Odd Integers

Quote:
If are odd integers, Prove that is odd iff k is odd.
Consider the sum:



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

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July 27th, 2012, 02:28 PM   #5
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Re: Sum of Odd Integers

Thanks for the responses, very helpful.
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July 27th, 2012, 03:29 PM   #6
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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.
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July 29th, 2012, 08:05 AM   #7
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Re: Sum of Odd Integers

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Originally Posted by HallsofIvy
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.
Isn't your method exactly same as I gave before?
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July 29th, 2012, 08:38 AM   #8
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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.
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July 31st, 2012, 01:09 AM   #9
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Re: Sum of Odd Integers

It is difficult for me to understand it.
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July 31st, 2012, 04:39 AM   #10
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Re: Sum of Odd Integers

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Originally Posted by jicheng
It is difficult for me to understand it.
What's difficult for you to understand? The solution I did?
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