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8. Sequences and Series
hard
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$\frac{5}{2}$
B$\frac{10}{9}$
C$\frac{3}{2}$
D$\frac{5}{3}$
(JEE MAIN-2025)
Solution
$S=\{0,1,2,3 \ldots \ldots\}$$\log _0 y=x \log _0\left(\frac{2}{5}\right)$
$\Rightarrow y=\left(\frac{2}{5}\right)^x$
image
Required
$\operatorname{Sum}=1+\left(\frac{2}{5}\right)^1+\left(\frac{2}{5}\right)^2+\left(\frac{2}{5}\right)^3+\ldots . .=\frac{1}{1-\frac{2}{5}}=\frac{5}{3}$
Standard 11
Mathematics
