Question

If $p,q,r,sϵN$ and they are four consecutive terms of an AP, then the $p$th, $q$th, $r$th, $s$th terms of a GP are in

• A. AP
• B. GP
• C. HP
• D. none of these

Answer 1

The correct option is B GP

$p,q,r,s$ are in AP
$\Rightarrow 2q=p+r$ and $2r=s+q$
$\Rightarrow 2r=s+\frac{p+r}{2}$
$\Rightarrow s=\frac{3r-p}{2}$
Now, the $p^{th},q^{th},r^{th},s^{th}$ terms of a GP can be written as
$a{t}^{p-1},a{t}^{q-1},a{t}^{r-1},a{t}^{s-1}$
$=a{t}^{p-1},a{t}^{\frac{p+r}{2}-1},a{t}^{r-1},a{t}^{s-1}$
$=a{t}^{p-1},a{t}^{p-1}{t}^{\frac{r-p}{2}},a{t}^{p-1}{t}^{r-p},a{t}^{\frac{3r-p}{2}-1}$
$=a{t}^{p-1},a{t}^{p-1}{t}^{\frac{r-p}{2}},a{t}^{p-1}{t}^{r-p},a{t}^{p-1}{t}^{\frac{3(r-p)}{2}}$
$=A,AR,A{R}^2,A{R}^3$ where $A=a{t}^{p-1}$ and $R=t^{\frac{r-p}{2}}$
Thus, the terms are in GP.
$\therefore$ Ans. is option B.