\begin{gather*} S_n = \sum_{k=0}^{n} {x^k} = 1 + x + x^2 + ... + x^n \\ (1-x)S_n = S_n - x S_n = 1 + x + x^2 + ... + x^n - (x + x^2 + ... + x^n + x^{n+1}) = 1 - x^{n+1} \\ \Rightarrow S_n = \frac{1-x^{n+1}}{1-x} \end{gather*}