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)

p, q, r are in A.P.

Answer 2

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