<s>[INST] Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$ 
Possible Duplicate:
Value of $\sum x^n$ 

Proof to the formula 
$$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$
 [/INST] Since $$\frac1{1-x}=1+x+x^2+x^3+\cdots,$$ we have
$$
\frac{1-x^{n+1}}{1-x}=(1+x+x^2+x^3+\cdots)-(x^{n+1}+x^{n+2}+x^{n+3}+x^{n+4}+\cdots)
$$
And on the right hand side every thing cancels except $1+x+x^2+\cdots+x^n$.
(This argument is probably circular! :) )
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