Let a 1 , a 2 , a 3 , . . . . . be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then a 1 a 9 + a 2 a 4 a 9 + a 5 + a 7 is equal to________ . [2023]

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Solution

Q.
Let $a_{1}, a_{2}, a_{3}, . . . . .$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then $a_{1} a_{9} + a_{2} a_{4} a_{9} + a_{5} + a_{7}$ is equal to________ . [2023]

Ans.
(60)

Given, $a_{4} \cdot a_{6} = 9$

$\Rightarrow {(a_{5})}^{2} = 9 \Rightarrow a_{5} = 3 and a_{5} + a_{7} = 24$

$\Rightarrow a_{5} + a_{5} r^{2} = 24 \Rightarrow a_{5} (1 + r^{2}) = 24$

$\Rightarrow (1 + r^{2}) = 8 \Rightarrow r = \sqrt{7} \Rightarrow a_{1} = \frac{3}{49}$

$So, a_{1} a_{9} + a_{2} a_{4} a_{9} + a_{5} + a_{7} = 9 + 27 + 3 + 21 = 60$

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