Question

The value of sum $\sum _{n=1}^{\infty }\frac{n}{7^n}$ is

• A. $\frac{5}{11}$
• B. $\frac{14}{9}$
• C. $\frac{4}{25}$
• D. $\frac{7}{36}$

Answer 1

The correct option is D $\frac{7}{36}$

Let $S=\sum _{n=1}^{\infty }\frac{n}{7^n}$
$\Rightarrow S=\frac{1}{7}+\frac{2}{7^2}+\frac{3}{7^3}+\frac{4}{7^4}+\dots$
$\Rightarrow S=\left(\frac{1}{7}\right)(1+\frac{2}{7}+\frac{3}{7^2}+\frac{4}{7^3}+\dots )$
Clearly, this is an infinite AGP with $a=1,r=\frac{1}{7},d=1$

$\Rightarrow S=\left(\frac{1}{7}\right)(\frac{a}{1-r}+\frac{dr}{(1-r{)}^2})$
$=\left(\frac{1}{7}\right)(\frac{1}{1-\frac{1}{7}}+\frac{1\times \frac{1}{7}}{{(1-\frac{1}{7})}^2})$
$=\frac{1}{7}(\frac{7}{6}+\frac{7}{36})$
$=\frac{7}{36}$