Question

In a GP, $(p+q)$ th term is m and $(p-q)$ th term is n, then the value of pth term is

• A. $\frac{m}{n}$
• B. $\sqrt{\frac{m}{n}}$
• C. $\sqrt{mn}$
• D. $\sqrt{\left(\frac{n}{m}\right)}$

Answer 1

Answer: (C)

In GP, $(p+q)th$ term $=m$
$\Rightarrow$ $a{r}^{p+q-1}=m$ ...(i)
and $(p-q)thterm=n$
$\Rightarrow$ $a{r}^{p-q-1}=n$ ...(ii)
Dividing Eq. (i) by Eq. (ii),
$\frac{a{r}^{p+q-1}}{a{r}^{p-q-1}}=\frac{m}{n}$
$\Rightarrow$ $r^{p+q-1-p+q+1}=\frac{m}{n}$
$\Rightarrow$ $r^{2q}=\frac{m}{n}$
$\Rightarrow$ $r={\left(\frac{m}{n}\right)}^{1/2q}$ ...(iii)
Now, pth term $=a{r}^{P-1}$
$=a{r}^{P+q-1}.r^{-q}$
$=m{\left[{\left(\frac{m}{n}\right)}^{1/2q}\right]}^{-q}$
[using Eqs. (i) and (iii)]
$=m{\left(\frac{m}{n}\right)}^{-1/2}$
$=m{\left(\frac{m}{n}\right)}^{1/2}=\sqrt{nm}$