Question

In a G.P. if the (m + n)^th term is p and (m − n)^th term is q, then its m^th term is
(a) 0
(b) pq
(c) $\sqrt{pq}$
(d) $\frac{1}{2}(p+q)$

Answer 1

(c) $\sqrt{pq}$

$$\begin{gathered} \mathrm{Here},a_{\left(m+n\right)}=p \\ \Rightarrow a{r}^{\left(m+n-1\right)}=p.......\left(\mathrm{i}\right) \\ \mathrm{Also},{\mathrm{a}}_{\left(m-n\right)}=q \\ \Rightarrow a{r}^{\left(m-n-1\right)}=q.......\left(\mathrm{ii}\right) \\ \mathrm{Mutliplying}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right): \\ \Rightarrow a{r}^{\left(m+n-1\right)}a{r}^{\left(m-n-1\right)}=pq \\ \Rightarrow a^2{r}^{\left(2m-2\right)}=pq \\ \Rightarrow {\left(a{r}^{\left(m-1\right)}\right)}^2=pq \\ \Rightarrow a{r}^{\left(m-1\right)}=\sqrt{pq} \\ \Rightarrow a_m=\sqrt{pq} \\ \mathrm{Thus},\mathrm{the}{m}^{\mathrm{th}}\mathrm{term}\mathrm{is}\sqrt{pq}. \end{gathered}$$