CAT 2024 Slot 2 QA Question 19: The sum of the infinite series $\dfrac{1}{5}\left(\dfrac{1}{5} - \dfrac{1}{7}\right) + \left(\dfrac{1}{5}\right)^2\left[\left(\dfrac{1}{5}\right)^2 - \left(\dfrac{1}{7}\right)^2\right] + \left(\dfrac{

Question

The sum of the infinite series $\dfrac{1}{5}\left(\dfrac{1}{5} - \dfrac{1}{7}\right) + \left(\dfrac{1}{5}\right)^2\left[\left(\dfrac{1}{5}\right)^2 - \left(\dfrac{1}{7}\right)^2\right] + \left(\dfrac{

Accepted Answer

The $n$th term of the series is:
$\left(\dfrac{1}{5}\right)^n\left[\left(\dfrac{1}{5}\right)^n - \left(\dfrac{1}{7}\right)^n\right]$
$= \left(\dfrac{1}{25}\right)^n - \left(\dfrac{1}{35}\right)^n$
So the sum is the difference of two infinite geometric series.
$S = \sum_{n=1}^{\infty}\left(\dfrac{1}{