JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let the first term a and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three is 33033, then the sum of these terms is equal to :

Choose the correct answer:

Explanation

Let the 3 terms of G.P are $a, ar, ar^2$ then according to the question, we have: $(a)^2 + (ar)^2 + (ar^2)^2 = 33033$ $\Rightarrow a^2 (1 + r^2 + r^4) = 11^2 \times (3 \times 7 \times 13)$ Since, $a$ and $r$ are positive integers, so by comparing we have $a = 11$ and $1 + r^2 + r^4 = 3 \times 7 \times 13$ . $\Rightarrow r^4 + r^2 - 272 = 0$ $\Rightarrow (r^2 - 16) (r^2 + 17) = 0$ $\Rightarrow r^2 = 16$ or $r^2 = -17$ (Not possible) $\Rightarrow r = 4$ ( $\because r$ is positive integer) So, the numbers are $11, 11 \times 4, 11 \times 4^2$ or $11, 44, 176$ . And sum of these numbers $= 11 + 44 + 176 = 231$ .