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Q.
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?
Sequences and Series
Solution:
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.