Question

If the $p^{th}$, $q^{th}$ and $r^{th}$ terms of a G.P. are again in G.P., then which one of the following is correct?

• A. p, q, r are in A.P.
• B. p, q, r are in G.P.
• C. p, q, r are in H.P.
• D. p, q, r are neither in A.P. nor in G.P. nor in H.P.

Answer 1

Answer: (A)

Let R be the common ratio of this GP and a be the first term. pth term is $a{R}^{p-1}$, qth term is $a{R}^{q-1}$ and rth term is $a{R}^{r-1}$.
Since p, q and r are in G.P. then $(a{R}^{q-1}{)}^2=a{R}^{p-1}.a{R}^{r-1}$
$\Rightarrow a^2{R}^{2q-2}=a^2{R}^{p+r-2}$
$\Rightarrow R^{2q-2}=R^{p+r-2}$
$\Rightarrow 2q-2=p+r-2$
$\Rightarrow 2q=p+r\Rightarrow p,q,r$ are in A.P.