Proof of Binomial Identity: Proving SUM(nCk)*2^k=(3^n+(-1)^n)/2

[proptrader]

Homework Statement

Prove that for all positive integers n, the equality holds:
SUM(nCk)*2^k=(3^n+(-1)^n)/2
Note: The sum goes from k=0 to n. AND k has to be even.

Homework Equations

Binomial Theorem

The Attempt at a Solution

I know that if we use the binomial theorem for x=2 and y=1, we would get
(2+1)^n=SUM(nCk)*2^k , with no restriction for k being even. I wonder how to proceed.
Thanks.

 

[LCKurtz]

Hint: Your idea looks like a good start. You have done (writing it in the other order) the expansion of (1 + 2)ⁿ.

Look at the same thing for (1 - 2)ⁿ and see if that gives you any ideas.