CAT 2025 Slot 2 QA Question 5: Let $a_n$ be the nth 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

Question

Let $a_n$ be the nth 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

Accepted Answer

Let the first term be $a$ and the common ratio be $r$. Since the infinite GP is decreasing, $0

$a(1+r+r^2) = 52$
$a^2r(1+r+r^2) = 624$
Dividing the second equation by the first, $ar = 12$

So the three terms can be written as
$\dfrac{12}{r}, 12, 12r$
$\dfrac{12}{r} + 12 + 12r = 52$
$\Rightarrow 3r

Let ana_nan​ be the nth term of a decreasing infinite geometric progression. If a1+a2+a3=52a_1 + a_2 + a_3 = 52a1​+a2​+a3​=52 and a1a2+a2a3+a3a1=624a_1 a_2 + a_2 a_3 + a_3 a_1 = 624a1​a2​+a2​a3​+a3​a1​=624, then the sum of this geometric progression is

Let $a_n$ be the n^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