How do you prove this?
Many thanks!
Let $\displaystyle S = 1 + x + x^2 + ... + x^N$
Then $\displaystyle xS =$ $\displaystyle x + x^2 + ....+ x^N + x^{N+1}$
Now subtract the two equations:
$\displaystyle S - xS = (1 + x + x^2+ ....+ x^N) - (x + x^2 + ....+ x^N + x^{N+1})$
Group terms now, so that they cancel,
$\displaystyle (1 - x)S = 1 +(x - x) + (x^2 - x^2)+ ....+(x^N - x^N) - x^{N+1}$
After canceling, we have:
$\displaystyle (1 - x)S = 1 - x^{N+1}$
Now since $\displaystyle x \neq 1$, divide by $\displaystyle (1 - x)$ on both sides and we are done!
NOTE: This is exactly what clic-clac did, but I have explicitly written down the terms so that you can "see" the solution clearly. Try reading through clic-clac's post, to learn elegant ways of writing proofs involving sums.