Geometric Series
#Math
$\displaystyle S_{n}=a\sum_{i = 0}^{n-1}r^{i}=a+ar+ar^{2}+... +ar^{n-1}=\frac{a(1-r^{n})}{1-r}$
- $\displaystyle S_{n}$ is the series sum
- $\displaystyle a$ is the initial value of the sequence
- $\displaystyle r$ is the ratio that the geometric series multiplies by
- $\displaystyle n$ is the number of terms in the series
Derivation
$$
\begin{align}
S_{n} &= a+ar+ar^{2}+\ldots +ar^{n-1} \ \\-,,,,,,,,,,S_{n}r &=0+ ar+ar^{2}+\ldots +ar^{n-1}+ar^{n} \\________&_______________________________\ \\S_{n}-S_{n}r&=a-ar^{n} \\S_{n}(1-r)&=a(1-r^{n}) \\S_{n}&=\frac{a(1-r^{n})}{1-r}
\end{align}
$$