Let S = N 0 . Define a relation R from S to R by : R = (x, y): _e… - Vidyadip

MCQ

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

A
$\frac{3}{2}$
✓
$\frac{5}{3}$
C
$\frac{10}{9}$
D
$\frac{5}{2}$

✓

Answer

Correct option: B.

$\frac{5}{3}$

(B) $\frac{5}{3}$
$\begin{array}{ll}\text {Sol. } S =\{0,1,2,3 \ldots . .\} \\ \log _{ c } y =\log _c\left(\frac{2}{5}\right) \\ \Rightarrow y =\left(\frac{2}{5}\right)^x\end{array}$

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}
$

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