# Question c2066

Mar 21, 2017

$\text{The Sum=} \frac{7776}{2401} \approx 3.23865 .$

#### Explanation:

$\text{The Sum=} {\sum}_{n = 5}^{\infty} \left({6}^{n} / {7}^{n}\right)$

$= {\sum}_{n = 5}^{\infty} {\left(\frac{6}{7}\right)}^{n}$

={(6/7)^5+(6/7)^6+(6/7)^7+..."(upto "oo)}#

$= {\left(\frac{6}{7}\right)}^{5} \left\{1 + \left(\frac{6}{7}\right) + {\left(\frac{6}{7}\right)}^{2} + \ldots \ldots\right\}$

Let us notice that the Series in $\left\{\ldots \ldots\right\}$ is a Geometric Series,

like, $\left\{a + a r + a {r}^{2} + \ldots \ldots\right\}$, for which, the Sum $s$ is given by,

$s = \frac{a}{1 - r} , \iff r < 1.$

Accordingly, in our Problem, $\because , r = \frac{6}{7} < 1 ,$

$\text{The Sum=} {\left(\frac{6}{7}\right)}^{5} \left\{\frac{1}{1 - \left(\frac{6}{7}\right)}\right\} = \left({6}^{5} / {7}^{5}\right) \left(7\right) = {6}^{5} / {7}^{4} , \mathmr{and} ,$

$\text{The Sum=} \frac{7776}{2401} \approx 3.23865 .$

Enjoy Maths.!