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Q.
Let $a_1, a_2, a_3, \ldots$ be a $G P$ 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 $a_1 a_9+a_2 a_4 \alpha_9+a_5+a_7$ is equal to ____
$a _4 \cdot a _6=9 \Rightarrow\left( a _5\right)^2=9 \Rightarrow a _5=3 $
$ \& a _5+ a _7=24 \Rightarrow a _5+ a _5 r ^2=24 \Rightarrow\left(1+ r ^2\right)=8 \Rightarrow r =\sqrt{7} $
$ \Rightarrow a =\frac{3}{49} $
$ \Rightarrow a _1 a _9+ a _2 a _4 a _9+ a _5+ a _7=9+27+3+21=60$