JEE 2025 — Mathematics PYQ

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Question

JEE 2025 — Mathematics PYQ

JEE | Mathematics | 2025

$Let a_{1}, a_{2}, a_{3}, \dots be a GP of increasing positive terms.$ $If a_{1} a_{5} = 28 and a_{2} + a_{4} = 29,$ $then a_{6} is equal to:$

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Explanation

a₁, a₂, ... GP

$a_{1} a_{5} = 28 and a_{2} + a_{4} = 29$

$Let a be the first term and r be the common ratio.$

$a (a r^{4}) = 28 and a r + a r^{3} = 29$

$a^{2} r^{4} = 28$

$a r^{2} = 27$

$\Rightarrow \frac{2 7}{r} + 27 r = 29$

$\Rightarrow 27 r^{2} - 29 r + 27 = 0$

$\Rightarrow 27 r^{2} - 28 r - r + 27 = 0$

$\Rightarrow 27 r (r - 27) - 1 (r - 27) = 0$

$\Rightarrow (r - 27) (27 r - 1) = 0$

$\Rightarrow r = 27, \frac{1}{2 7}$

$∵ Increasing GP \Rightarrow r = 27$

$∴ a = \frac{2}{2 7}$

$∴ a_{6} = a r^{5} = (27)^{4} = (28)^{2} = 784$

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