# How do you find the sum of the infinite geometric series 18,12,8,...?

Apr 15, 2016

Sum of the infinite geometric series $\left\{18 , 12 , 8 , . .\right\}$ is $54$

#### Explanation:

Sum ${S}_{n}$ of a geometric series $\left\{a , a r , a {r}^{2} , a {r}^{3} , a {r}^{4} , \ldots .\right\}$ upto $n$ terms, whose first term is $a$ and ratio of a term to its preceding term is $r$ is given by

$a \frac{{r}^{n} - 1}{r - 1}$, when $r > 1$

or $a \frac{1 - {r}^{n}}{1 - r}$ when $r < 1$.

When $n \to \infty$, $L t {S}_{n} \to \frac{a}{1 - r}$

Here in the series $\left\{18 , 12 , 8 , . .\right\}$ $r = \frac{12}{18} = \frac{8}{12} = \frac{2}{3} < 1$

Hence when $n \to \infty$, $L t {S}_{n} \to \frac{18}{1 - \frac{2}{3}} = \frac{18}{\frac{1}{3}} = 18 \times \frac{3}{1} = 54$

Hence, sum of the infinite geometric series $\left\{18 , 12 , 8 , . .\right\}$ is $54$.