Question

Evaluate $1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+.......$ to infinite series

• A. $\frac{25}{8}$
• B. $\frac{8}{25}$
• C. $3$
• D. None of these

Answer 1

The correct option is D None of these

Series:

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

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

$\therefore$ eq (1) - eq(2)

$\Rightarrow S(1-\frac{1}{5})=1+\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+...\infty$

$\Rightarrow S\left(\frac{4}{5}\right)=1+3.\frac{\frac{1}{5}}{1-\frac{1}{5}}$

$\Rightarrow S\left(\frac{4}{5}\right)=\frac{7}{4}$

$\therefore S=\frac{35}{16}$