Question

The sum of infinity 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. $\frac{8}{6}$

Answer 1

The correct option is C $\frac{35}{16}$

Let $S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+...$ ...(1)
Multiply (1) by $\frac{1}{5}$ , we get
$\frac{1}{5}S=\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+\frac{10}{5^4}+...$ ...(2)
$\left(1\right)-\left(2\right)$, gives
$(1-\frac{1}{5})S=1+\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+...$
$\Rightarrow \frac{4}{5}S=1+\frac{3}{5}(1+\frac{1}{5}+\frac{1}{5^2}+...)$
$\Rightarrow \frac{4}{5}S=1+\frac{3}{5}\left(\frac{1}{1-\frac{1}{5}}\right)\Rightarrow \frac{4}{5}S=1+\frac{3}{5}\left(\frac{5}{4}\right)$
$\Rightarrow \frac{4}{5}S=1+\frac{3}{4}\Rightarrow S=\frac{35}{16}$