Question

The $(m+n)$ th and $(m-n)$ th terms of a $G.P$. are $p$ and $q$ respectively. Then the $m$ th term of the $G.P$. is

• A. $p{\left(\frac{q}{p}\right)}^{m/2n}$
• B. $\sqrt{pq}$
• C. $\sqrt{p/q}$
• D. None of these

Answer 1

Answer: (B)

Let $a$ be the first term and $r$ be the common ratio.
Then, $a_{m+n}=p$ and $a_{m-n}=q$
$\Rightarrow a{r}^{m+n-1}=p\text{ and }a{r}^{m-n-1}=q$
$\Rightarrow \left(a{r}^{m+n-1}\right)\left(a{r}^{m-n-1}\right)=pq$
$\Rightarrow a^2{r}^{2m-2}=pq$
$\Rightarrow a{r}^{m-1}=\sqrt{pq}$
$\Rightarrow a_m=m\text{ th term }=\sqrt{pq}$