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 __________.
Answer (integer)
60
Solution
Let $r$ be the common ratio of the G.P
<br/><br/>
$\therefore a_{1} r^{3} \times a_{1} r^{5}=9$
<br/><br/>
$a_{1}^{2} r^{8}=9 \Rightarrow a_{1} r^{4}=3$
<br/><br/>
And
<br/><br/>
$$
\begin{aligned}
& a_{1}\left(r^{4}+r^{6}\right)=24 \\\\
\Rightarrow & 3\left(1+r^{2}\right)=24 \\\\
\therefore & r^{2}=7 \text { and } a_{1}=\frac{3}{49}
\end{aligned}
$$
<br/><br/>
Now
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$$
\begin{aligned}
& a_{1} a_{9}+a_{2} a_{4} a_{9}+a_{5}+a_{7} \\\\
& =a_{1}^{2} r^{8}+a_{1}^{3} r^{12}+24 \\\\
& =24+\frac{9}{7^{4}} \times 7^{4}+\frac{27}{7^{6}} \cdot 7^{6}=60
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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