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How do I use pascals identity:

$${2n\choose 2k}={2n-1\choose 2k}+{2n-1\choose 2k-1}$$ to prove that

$$\displaystyle\sum_{k=0}^{n}{2n\choose 2k}=\displaystyle\sum_{k=0}^{2n-1}{2n-1\choose k}$$

for every positive integer $n$ ?

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    Try writing out Pascal's identity for n = 1, 2, 3, 4 and writing out the equation that you're trying to prove for n = 4 (without simplifying the binomial coefficients). See if you can give a proof in that case. If you're able to do this, you'll see how to do the general case.2012-10-26

4 Answers 4