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Binomial Identity

  1. Sep 17, 2011 #1
    1. The problem statement, all variables and given/known data
    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.


    2. Relevant equations
    Binomial Theorem


    3. 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.
     
  2. jcsd
  3. Sep 17, 2011 #2

    LCKurtz

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    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.
     
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