Let S N ∪ {0}. Define a relation R from S to R by: R=(x,y):loge⁡y=xloge⁡25,x∈S,y∈R. Then, the sum of all the elements in the range of R is equal to

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

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answer is B.

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Detailed Solution

S = {0, 1, 2, 3 …..}
$\begin{array}{l} \log_{0} y = {xlog}_{0} (\frac{2}{5}) \\ \Rightarrow y = {(\frac{2}{5})}^{x} \end{array}$
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Sum = $1 + {(\frac{2}{5})}^{1} + {(\frac{2}{5})}^{2} + {(\frac{2}{5})}^{3} + \dots . - = \frac{1}{1 - \frac{2}{5}} = \frac{5}{3}$