Question

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

• A. $\frac{7}{15}$
• B. $\frac{35}{16}$
• C. $\frac{41}{19}$
• D. None of these

Answer 1

Answer: (B)

$\frac{35}{16}$

Here, the series is an A.G.P., where $1,4,7,10,..$ is an A.P., with $a=1,d=3$ and $1,\frac{1}{5},\frac{1}{5^2},...$ is a G.P. with $r=\frac{1}{5}$.

For an A.G.P. $S_{\infty }=\frac{a}{1-r}+\frac{rd}{(1-r{)}^2}$

$S_{\infty }=\frac{1}{1-\frac{1}{5}}+\frac{\frac{1}{5}\cdot 3}{{(1-\frac{1}{5})}^2}=\frac{5}{4}+\frac{15}{16}=\frac{35}{16}$

By putting the values, we get $S_{\infty }=\frac{35}{16}$