Question

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

• A. $\frac{7}{16}$
• B. $\frac{5}{16}$
• C. $\frac{105}{64}$
• D. $\frac{35}{16}$

Answer 1

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

$S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+...$

$-\frac{s}{3}=\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+...$

___________________________

$\frac{4s}{5}=1+\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+...$

$=1+\frac{3}{5}(1+\frac{1}{5}+\frac{1}{5^2}+...)$

$\{a+a{r}^2+a{r}^3+...=\frac{a}{1-r}if|r|<1\}$

$=1+\frac{3}{5}\left(\frac{1}{1-1/5}\right)$

$\frac{4s}{5}=1+\frac{3}{5}.\frac{5}{4}=1+\frac{3}{4}=\frac{7}{4}$

$\Rightarrow s=\frac{35}{16}$