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# The first term of a $GP$ is $4$ and the sum of its $3^{rd}$ and $5^{th}$ terms is $360$. Find the common ratio.

$\begin{array}{1 1} 10 \\ 9 \\ 3 \\ 4 \end{array}$

$ar^2\;+\;ar^4=360$
$r^4+r^2=90$
$r^4+r^2-90=0$
$r^4+10r^2-9r^2-90=0$
$r^2(r^2+10)-9(r^2+10)=0$
$(r^2-9)\;(r^2+10)=0$
$Since\quad\;r^2\;\neq\;-10$
$r^2-9=0\quad\;r^2=9\quad\;r=3.$