Question

In a G.P. if $(m+n{)}^{th}$ term is $p$ and $(m-n{)}^{th}$ term is $q$, then $m^{th}$ term is:

• A. $\frac{p}{q}$
• B. $\frac{q}{p}$
• C. $pq$
• D. $\sqrt{pq}$

Answer 1

Answer: (D)

For a G. P , $a_{m+n}=p$ and $a_{m-n}=q$,
We know that $a_n=A{R}^{n-1}$ (in G.P.)
where A= first term and $R=$ ratio
$\therefore a_{m+n}=p$
$\Rightarrow A{R}^{m+n-1}=p\dots (i)$
and $a_{m-n}=q$
$\Rightarrow A{R}^{m-n-1}=q\dots (ii)$
On multiply equations (i) and (ii), we have
$\left(A{R}^{m+n-1}\right)\cdot \left(A{R}^{m-n-1}\right)=pq$
$\Rightarrow A^2\cdot R^{2(m-1)}=pq$
$\Rightarrow {\left(A{R}^{m-1}\right)}^2=pq$
$\Rightarrow A{R}^{m-1}=\sqrt{pq}$
$\Rightarrow a_m=\sqrt{pq}$