# How do you find the sum of the finite geometric sequence of sum_(i=0)^10 5(-1/3)^(i-1)?

May 30, 2018

$S = 5 {\left(- \frac{1}{3}\right)}^{- 1} \left(\frac{1 - {\left(- \frac{1}{3}\right)}^{11}}{1 - \left(- \frac{1}{3}\right)}\right) = - \frac{221435}{19683}$

#### Explanation:

The $n$ term finite geometric series that starts with 1 has a straightforward sum:

${S}_{n} = {\sum}_{k = 0}^{n - 1} {r}^{k} = \frac{1 - {r}^{n}}{1 - r}$

For the general geometric series we multiply by the first term. Our series has 11 terms.

$S = {\sum}_{i = 0}^{10} 5 {\left(- \frac{1}{3}\right)}^{i - 1}$

$S = 5 {\left(- \frac{1}{3}\right)}^{- 1} \left(\frac{1 - {\left(- \frac{1}{3}\right)}^{11}}{1 - \left(- \frac{1}{3}\right)}\right)$

$S = - \frac{221435}{19683}$