CAT 2025 Slot 2 QA Q8 Progressions & Series

Question 8 of the Quantitative Ability (QA) section, CAT 2025 Slot 2, with CAT's official answer and a worked solution.

Let \( a _ { n } \) be the \( n ^ { \text {th } } \) term of a decreasing infinite geometric progression. If \( a _ { 1 } + a _ { 2 } + a _ { 3 } = 52 \) and \( a _ { 1 } a _ { 2 } + a _ { 2 } a _ { 3 } + a _ { 3 } a _ { 1 } = 624 \), then the sum of this geometric progression is

A
\( 63 \)
B
\( 60 \)
C
\( 54 \)
D
\( 57 \)

Answer: (C) 54 Progressions & Series MCQ

Solution

Terms \(a, ar, ar^2\): \(a(1 + r + r^2) = 52\).
\(a_1a_2 + a_2a_3 + a_3a_1 = a^2r(1 + r + r^2) = ar \cdot 52 = 624 \Rightarrow ar = 12\).
Then \(12\left(\frac1r + 1 + r\right) = 52 \Rightarrow r + \frac1r = \frac{10}{3} \Rightarrow r = \frac13\) (decreasing).
\(a = 36\), sum \(= \frac{36}{1 - \frac13} = 54\).

Shortcut. Terms are 36, 12, 4: check 52 and \(432 + 48 + 144 = 624\).