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How do I solve this summation?

I have tried simplifying k on the numerator with the factorial in the denominator but it just gets me nowhere.



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up vote 7 down vote accepted



You get a telescopic sum.

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I thought that maybe you could form a telescopic sum but I couldn't figure it out! Thanks! – Csmity Nov 23 '13 at 19:39
@Csmity The reason why I thought about this solution is that you had $k+2$ in the denominator and $k$ at the top. Their difference is exactly the $2$ in $2^k$, thus it was natural to write $k+2-k=2$ or $k=(k+2)-2$ and see what we get.... – N. S. Nov 23 '13 at 19:41

Using the Hint given by N.S.:


Using telescopic sum we get:


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