# How do you find the sum of the infinite geometric series, a(1)= -5, r= 1/6?

Mar 24, 2016

Sum of infinite series is $- 6$

#### Explanation:

Sum of geometric series $\left\{a , a r , a {r}^{2} , a {r}^{3} , , , , , , , , , ,\right\}$ up to $n$ terms is given by

$S = a \times \frac{{r}^{n} - 1}{r - 1}$ if $r > 0$ and

$S = a \times \frac{1 - {r}^{n}}{1 - r}$ if $r < 0$

If $r < 0$, $L {t}_{n \to \infty} \left({r}^{n}\right) = 0$ and hence $S = \frac{a}{1 - r}$

Here as $a = - 5$ and $r = \frac{1}{6}$ and as $r < 1$

$S = - \frac{5}{1 - \frac{1}{6}} = - \frac{5}{\frac{5}{6}} = - 5 \times \frac{6}{5} = - 6$