CAT 2023 Slot 3 QA Question 20: The value of $1 + \left(1 + \dfrac{1}{3}\right)\dfrac{1}{4} + \left(1 + \dfrac{1}{3} + \dfrac{1}{9}\right)\dfrac{1}{16} + \left(1 + \dfrac{1}{3} + \dfrac{1}{9} + \dfrac{1}{27}\right)\dfrac{1}{64} + \c

Question

The value of $1 + \left(1 + \dfrac{1}{3}\right)\dfrac{1}{4} + \left(1 + \dfrac{1}{3} + \dfrac{1}{9}\right)\dfrac{1}{16} + \left(1 + \dfrac{1}{3} + \dfrac{1}{9} + \dfrac{1}{27}\right)\dfrac{1}{64} + \c

Accepted Answer

The term $\dfrac{1}{3^j}$ first appears with multiplier $\dfrac{1}{4^j}$ and then continues in every later term.

So its total multiplier is

$\dfrac{1}{4^j} + \dfrac{1}{4^{j + 1}} + \dfrac{1}{4^{j + 2}} + \cdots$
$= \dfrac{\dfrac{1}{4^j}}{1 - \dfrac{1}{4}}$
$= \dfrac{4}{3 	imes 4^j}$

Hence, the t