Let $S=\mathbf{N} \cup\{0\}$. Define a relation $R$ from $S$ to $\mathbf{R}$ by : $\mathrm{R}=\left\{(x, y): \log _{\mathrm{e}} y=x \log _{\mathrm{e}}\left(\frac{2}{5}\right), x \in \mathrm{~S}, y \in \mathbf{R}\right\}$ Then, the sum of all the elements in the range of $R$ is equal to :

• A

  $\frac{10}{9}$
• B

  $\frac{3}{2}$
• C

  $\frac{5}{2}$
• Correct answer
  D

  $\frac{5}{3}$

Solution

$\begin{aligned} & S=\{0,1,2,3 \ldots \ldots\} \\ & \log _y y=\log _e\left(\frac{2}{5}\right) \\ & \Rightarrow y=\left(\frac{2}{5}\right)^x\end{aligned}$
Required $\text { Sum }=1+\left(\frac{2}{5}\right)^1+\left(\frac{2}{5}\right)^2+\left(\frac{2}{5}\right)^3+\ldots . .-=\frac{1}{1-\frac{2}{5}}=\frac{5}{3}$

Hence, the correct answer is option: 'D'.