# Geometric Series ![[Geometric Sequence#^ep8vl9]] The sum of first $n$ terms of a geometric sequence is: $\sum_{k=0}^n r^k = \frac{r^{n+1}-1}{r-1} \text{, for $r \ne 1$}$ If $r=1$ then the sum is $n+1$, as we're just adding 1 $n+1$ times. ^4zycdx For infinite series see [[Geometric Infinite Series]]. ## Proof $ \begin{align} S_n &= r^0 + r^1 + r^2 + \ldots + r^{n-1} + r^n && (1) \\ rS_n &= r^1 + r^2 + r^3 + \ldots + r^n + r^{n + 1} && (2) \\ rS_n - S_n &= r^{n+1} - 1 && (2) - (1) \\ S_n(r-1) &= r^{n+1}-1 \\ S_n &=\frac{r^{n+1} -1}{r-1} \end{align} $ For details see [Geometric Progression](https://brilliant.org/wiki/geometric-progressions) from Brilliant.