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)th\,term=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} $