Question

The sum of the infinite terms of the series $1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+...$ is

• A. $\frac{16}{35}$
• B. $\frac{11}{8}$
• C. $\frac{35}{16}$
• D. $\frac{9}{16}$

Answer 1

Answer: (C)

Let $S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+....$
$\frac{1}{5}S=\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+......$
On subtracting, $\left(S-\frac{1}{5}S\right)=1+\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+....$
$\Rightarrow$ $\frac{4}{5}S=1+\frac{3}{5}\left[1+\frac{1}{5}+\frac{1}{5^2}+.......\right]$
$=1+\frac{3}{5}\left[\frac{1}{1-1/5}\right]=1+\frac{3}{5}\times \frac{5}{4}$
$\Rightarrow$ $\frac{4}{5}S=1+\frac{3}{4}=\frac{7}{4}$
$\Rightarrow$ $S=\frac{7}{4}\times \frac{5}{4}=\frac{35}{16}$