Mathematical Induction: Prove P(n)=(n^2)-n divisible by 2

[portia-h]

Can you help me prove that
P(n) = (n^2) - n is divisible by 2

by using a P(k+1) - P(k) method, I have arrived at

2B - 2k = 2A and 2k^2 - 2B = 2A

but i don't think this is right.
quick help would be appreciated
thanks!

 

[pka]

Re: Mathematical Induction

Say that we know that $\displaystyle K^2-K$ is divisible by 2.
Consider $\displaystyle (K+1)^2-(K+1)=K^2+2K+1-K-1=K^2-K +2K$
Do you see how to finish?

 

[galactus]

Re: Mathematical Induction

First, show n=1 is true. Then we assume 2 is a factor of $\displaystyle k^{2}-k$

equivalently, $\displaystyle k^{2}-k=2p$ for some integer p.

We want to show that $\displaystyle P_{k+1}$ is true. Namely, 2 is a factor of:

$\displaystyle (k+1)^{2}-(k+1)$

$\displaystyle =k^{2}+2k+1-k-1$

Rearrange terms:

$\displaystyle =\underbrace{(k^{2}-k)}_{\text{2p}}+2k$

Now, can you finish?.

 

[portia-h]

Re: Mathematical Induction

i understand it completely now
thanks so much