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QJEEM23Ma7716: JEE Main 2023 Mathematics - Class 11 JEEM23Ma7716 Solution: Let be a of increasing positive numbers. If the product of…

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 $a_1a_9+a_2a_4a_9+a_5+a_7$ is equal to __________.

Step-by-Step Verified Solution

Let

r

be the common ratio of the G.P

\therefore a_{1} r^{3} \times a_{1} r^{5}=9

a_{1}^{2} r^{8}=9 \Rightarrow a_{1} r^{4}=3

And

\begin{aligned}&amp; a_{1}\left(r^{4}+r^{6}\right)=24 \\\\\Rightarrow &amp; 3\left(1+r^{2}\right)=24 \\\\\therefore &amp; r^{2}=7 \text { and } a_{1}=\frac{3}{49}\end{aligned}

Now

\begin{aligned}&amp; a_{1} a_{9}+a_{2} a_{4} a_{9}+a_{5}+a_{7} \\\\&amp; =a_{1}^{2} r^{8}+a_{1}^{3} r^{12}+24 \\\\&amp; =24+\frac{9}{7^{4}} \times 7^{4}+\frac{27}{7^{6}} \cdot 7^{6}=60\end{aligned}

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