Let S=N 0.

Question

Let $S=N\cup {0}$. Define a relation R from S to $R$ by :

$R={(x,y):{\log }_{e}y=x{\log }_{e}(\frac{2}{5}),x\in S,y\in R}.$

Then, the sum of all the elements in the range of $R$ is equal to :

Accepted Answer

$$\begin{aligned}
& S=\{0,1,2,3 \ldots . .\} \
& \log _{\mathrm{e}} \mathrm{y}=\log _{\mathrm{e}}\left(\frac{2}{5}\right) \
& \Rightarrow \mathrm{y}=\left(\frac{2}{5}\right)^{\mathrm{x}}
\end{aligned}$$

Required

$$\text { Sum }=1+\left(\frac{2}{5}\right)^1+\left(\frac{2}{5}\right)^2+\left(\frac{2}{5}\right)^3+\ldots . .-=\frac{1}{1-\frac{2}{5}}=\frac{5}{3}$$

Solutions · 1