Question

Sum of infinity of the series $1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+\cdots$ is

• A. $\frac{7}{16}$
• B. $\frac{5}{16}$
• C. $\frac{104}{64}$
• D. $\frac{35}{16}$

Answer 1

The correct option is D

$\frac{35}{16}$

Explanation for the correct option :

Step-1 : Assumption and some modification

Let us consider $S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+\cdots$ $\left(1\right)$

Dividing both sides of $\left(1\right)$ by $5$, we get :

$\frac{S}{5}=\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+\frac{10}{5^4}+\cdots$ $\left(2\right)$

Now, subtracting $\left(2\right)$ from $\left(1\right)$, we get :

$$\begin{gathered} S-\frac{S}{5}=\left(1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+\cdots \right)-\left(\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+\frac{10}{5^4}+\cdots \right) \\ \Rightarrow \frac{4S}{5}=1+\left(\frac{4}{5}-\frac{1}{5}\right)+\left(\frac{7}{5^2}-\frac{4}{5^2}\right)+\left(\frac{10}{5^3}-\frac{7}{5^3}\right)+\cdots \\ \Rightarrow \frac{4S}{5}=1+\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+\cdots \end{gathered}$$

Step-2 : Finding the value of $\frac{4S}{5}$

Now the R.H.S. of the above equation can be written as $1+\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}+\cdots =1+3\left(\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+\cdots \right)$.

The series $\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+\cdots$ is an infinite geometric series with the first term $\frac{1}{5}$ and common ratio $\frac{1}{5}$ .

We know that the sum of an infinite geometric series with the first term $a$ and common ratio $r$ is $\frac{a}{1-r}$.

So, the sum of $\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+\cdots$ will be

$$\begin{gathered} \frac{{\displaystyle \frac{1}{5}}}{1-{\displaystyle \frac{1}{5}}} \\ =\frac{{\displaystyle \frac{1}{5}}}{{\displaystyle \frac{4}{5}}} \\ =\frac{1}{4} \end{gathered}$$

Hence,

$$\begin{gathered} \frac{4S}{5}=1+3\left(\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+\cdots \right) \\ =1+3\times \frac{1}{4} \\ =\frac{7}{4} \end{gathered}$$

Step-3 : Finding the value of $S$

We have

$$\begin{gathered} \frac{4S}{5}=\frac{7}{4} \\ \Rightarrow S=\frac{7\times 5}{4\times 4} \\ \therefore S=\frac{35}{16} \end{gathered}$$

Hence, option (D) is the correct answer.