Geometric Series

$S_n=a\sum _{i=0}^{n-1}{r}^i=a+ar+a{r}^2+...+a{r}^{n-1}=\frac{a(1-r^n)}{1-r}$

• $S_n$ is the series sum
• $a$ is the initial value of the sequence
• $r$ is the ratio that the geometric series multiplies by
• $n$ is the number of terms in the series

Derivation

$$\begin{array}{rl}{S}_n& =a+ar+a{r}^2+\dots +a{r}^{n-1}\\ & \\ -S_{n}r& =0+ar+a{r}^2+\dots +a{r}^{n-1}+a{r}^n\\ \_\_\_\_\_\_\_\_& \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\ S_n-S_{n}r& =a-a{r}^n\\ S_n(1-r)& =a(1-r^n)\\ S_n& =\frac{a(1-r^n)}{1-r}\end{array}$$

Applications

• Compounding Interest