<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] For a more mechanical proof, you could use induction. The proof then boils down to finding a common denominator:
$\frac{x^{n+1}-1}{x-1} + x^{n+1} = \frac{x^{n+1}-1+(x-1)x^{n+1}}{x-1} = \frac{x^{n+2}-1}{x-1}$
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