CAT 2023 Slot 3 QA Q20 Progressions & Series

Question 20 of the Quantitative Ability (QA) section, CAT 2023 Slot 3, with CAT's official answer and a worked solution.

The value of \(1+\left(1+\frac{1}{3}\right) \frac{1}{4}+\left(1+\frac{1}{3}+\frac{1}{9}\right) \frac{1}{16}+\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}\right) \frac{1}{64}+\cdots\), is

A
\(\frac{15}{13}\)
B
\(\frac{16}{11}\)
C
\(\frac{27}{12}\)
D
\(\frac{15}{8}\)

Answer: (B) 16/11 Progressions & Series MCQ

Solution

The k-th term (from k = 0) is \(\left(\sum_{j=0}^{k}3^{-j}\right)4^{-k} = \frac32\left(1-3^{-(k+1)}\right)4^{-k}\).
Sum \(= \frac32\left[\sum 4^{-k} - \frac13\sum 12^{-k}\right] = \frac32\left[\frac43 - \frac13\cdot\frac{12}{11}\right]\).
\(= \frac32\left[\frac43-\frac4{11}\right] = \frac32\cdot\frac{32}{33} = \frac{16}{11}\).