Proving Expressions

johnny101

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Anyone have any idea how to proceed with this?Show that the sum of an odd number of integers is odd and then deduce that for any positive integer n, if a sum of n odd integers is odd, then n is even.
 
Anyone have any idea how to proceed with this?Show that the sum of an odd number of integers is odd and then deduce that for any positive integer n, if a sum of n odd integers is odd, then n is even.
You have a fundamental difficulty here: you cannot prove something that isn't true!.
For example, 2, 3, and 5 are an odd number of integers but their sum, 2+ 3+ 5= 10, is NOT odd. \

Did you mean "the sum of an odd number of odd integers"? Every odd number is of the form \(\displaystyle 2n_k+ 1\) for some integer \(\displaystyle n_k\). Do you see that if you want to add a list such odd numbers you could add all of the "\(\displaystyle 2n_k\)" together and then add the "1"s? Do you see that the sum of the \(\displaystyle 2n_k\) terms will be positive? What about a sum of "1"s? When will that be even and when will it be odd?
 
The question is what the text says, so I went with that. As for what you stated, can you elaborate on that with some example values? I'm not sure I'm understanding the latter half of the expression and what you're saying.
 
The question is what the text says, so I went with that. As for what you stated, can you elaborate on that with some example values? I'm not sure I'm understanding the latter half of the expression and what you're saying.
Halls gave you an example. The sum of any number, odd or even, of even integers is even. Furthermore, the sum of any even number of odd numbers plus any number of even numbers is even. So obviously the sum of three even numbers is even as is the sum of two odd numbers and one even number.

2 + 8 + 12 = 22.

3 + 7 + 2 = 12.

And you can't prove what isn't true.

It is, however, possible to prove that the sum of an odd number of odd integers is odd. But that does not seem to be your problem.
 
It is, however, possible to prove that the sum of an odd number of odd integers is odd. But that does not seem to be your problem.

For argument's sake, how would you prove the sum of an odd number of odd integers? I understand what you provided above now thanks!
 
So by showing both formulas for the odd/even it satisfies both aspects of the original question then?
 
So by showing both formulas for the odd/even it satisfies both aspects of the original question then?
Whoa. We cannot help you prove what you have repeatedly told us is the original proposition because it is false.
 
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