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]

Question

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]

Smart Achievers · Accepted Answer

(60)

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

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

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

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

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

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]

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Solution

Q. Let a1,a2,a3,..... 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 a1a9+a2a4a9+a5+a7 is equal to________ . [2023]

Ans. (60) Given, a4·a6=9 ⇒(a5)2=9⇒a5=3 and a5+a7=24 ⇒a5+a5r2=24⇒a5(1+r2)=24 ⇒(1+r2)=8⇒r=7⇒a1=349 So, a1a9+a2a4a9+a5+a7=9+27+3+21=60

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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]