# Prove that 2n^2 + n is odd if and only if cos(npi/2) is even

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• Oct 2nd 2013, 03:56 PM
MadSoulz
Prove that 2n^2 + n is odd if and only if cos(npi/2) is even
Let n be an element of Z. Prove that $2n^2 + 2$ is odd if and only if $cos\left(\frac{\pi n }{2}\right)$ is even.

Since it's a biconditional, I'm certain there will be two parts: 1) Proving $p \Rightarrow q$ , and 2) proving $q \Rightarrow p$

It's in the proof by contrapositive section, so I'll try to prove it that way.

1) If $2n^2 + 2$ is odd, then $cos\left(\frac{\pi n }{2}\right)$ is even.

Contrapositive: If $cos\left(\frac{\pi n }{2}\right)$ is odd, then $2n^2 + 2$ is even.

Assuming $cos\left(\frac{\pi n }{2}\right)$ is odd, then $n=2k$

$2n^2 + 2=2(2k)^2+2$

$8k^2+2=2(4k^2+1)=2m$ for some integer m.

Therefore, $2n^2 + 2$ is even and the implication is true.

My professor says that my proof is wrong, but I don't see how.
• Oct 2nd 2013, 04:10 PM
HallsofIvy
Re: Prove that 2n^2 + n is odd if and only if cos(npi/2) is even
I would suggest you go back and reread the problem. $2n^2+ 2= 2(n^2+ 1)$ is never odd.

Your title, however, says $2n^2+ n$, not $2n^2+ 2$. Perhaps if you tried proving that you will do better.
• Oct 2nd 2013, 04:25 PM
MadSoulz
Re: Prove that 2n^2 + n is odd if and only if cos(npi/2) is even
Quote:

Originally Posted by HallsofIvy
I would suggest you go back and reread the problem. $2n^2+ 2= 2(n^2+ 1)$ is never odd.

Your title, however, says $2n^2+ n$, not $2n^2+ 2$. Perhaps if you tried proving that you will do better.

Sorry, I have no idea why I typed '2' instead of 'n'
• Oct 2nd 2013, 04:28 PM
MadSoulz
Re: Prove that 2n^2 + n is odd if and only if cos(npi/2) is even
Correction: Let n be an element of Z. Prove that $2n^2 + n$ is odd if and only if $cos\left(\frac{\pi n }{2}\right)$ is even.

1) If $2n^2 + n$ is odd, then $cos\left(\frac{\pi n }{2}\right)$ is even.

Contrapositive: If $cos\left(\frac{\pi n }{2}\right)$ is odd, then $2n^2 + n$ is even.

Assuming $cos\left(\frac{\pi n }{2}\right)$ is odd, then $n=2k$

$2n^2 + n=2(2k)^2+2k$

$8k^2+2k=2(4k^2+k)=2m$ for some integer m.

Therefore, $2n^2 + n$ is even and the implication is true.