# Deriving closed formulas for summations [duplicate]

Possible Duplicate:
Value of $\sum x^n$

I was wondering how to derive a closed formula for things like $\sum_{i=1}^{n}2^{n}$=$2(2^{n}-1)$ and $\sum_{n=k}^{n}2^{n-k}$=$2^{n-k+1}-1$. I haven't done this in a while, and had wolfram do it for me, and I am not sure what the general tactic in getting these formulas is. Your help is appreciated!

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## marked as duplicate by Aryabhata, Ross Millikan, Mike Spivey, Akhil MathewApr 6 '11 at 4:56

Well, you can consider three ways, for one, use the inductive method; for two, try to write down what you want to sum, and then multiply it by 2 and subtract them; for three, try to write it as a recurrence relation, and compute it as usual. In each case, just remember that x-1 is a divisor of $x^n -1$ when n is a strictly positive natural number.