The $5^{\text {th }}, 8^{\text {th }}$ and $11^{\text {th }}$ terms of a $G.P.$ are $p, q$ and $s,$ respectively. Show that $q^{2}=p s$
Let $a$ be the first term and $r$ be the common ratio of the $G.P.$ According to the given condition,
$a_{5}=a r^{5-1}=a r^{4}=p$ .........$(1)$
$a_{8}=a r^{8-1}=a r^{7}=q$ .........$(2)$
$a_{11}=a r^{11-1}=a r^{10}=s$ .........$(3)$
Dividing equation $(2)$ by $(1),$ we obtain
$\frac{a r^{7}}{a r^{4}}=\frac{q}{p}$
$r^{3}=\frac{q}{p}$ .........$(4)$
Dividing equation $(3)$ by $(2),$ we obtain
$\frac{a r^{10}}{a r^{7}}=\frac{s}{q}$
$\Rightarrow r^{3}=\frac{s}{q}$ .......$(5)$
Equating the values of $r^{3}$ obtained in $(4)$ and $(5),$ we obtain
$\frac{q}{p}=\frac{s}{q}$
$\Rightarrow q^{2}=p s$
Thus, the given result is proved.
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