# How do you find the n-th partial sum of a geometric series?

Oct 22, 2014

Let us find a formula for the nth partial sum of a geometric series.

${S}_{n} = a + a r + a {r}^{2} + \cdots + a {r}^{n - 1}$

by multiplying by $r$,

$R i g h t a r r o w r {S}_{n} = a r + a {r}^{2} + \cdots + a {r}^{n - 1} + a {r}^{n}$

by subtracting $r {S}_{n}$ from ${S}_{n}$,

$R i g h t a r r o w \left(1 - r\right) {S}_{n} = a - a {r}^{n} = a \left(1 - {r}^{n}\right)$

(Notice that all intermediate terms are cancelled out.)

by dividing by $\left(1 - r\right)$,

$R i g h t a r r o w {S}_{n} = \frac{a \left(1 - {r}^{n}\right)}{1 - r}$

I hope that this was helpful.