Question

If in a $GP$ the $(p+q{)}^{th}$ term is $a$ and the $(p-q{)}^{th}$ term is $b,$ then $p^{th}$ term is

• A. ${\left(ab\right)}^{1/3}$
• B. ${\left(ab\right)}^{1/2}$
• C. ${\left(ab\right)}^{1/4}$
• D. None of these

Answer 1

The correct option is B ${\left(ab\right)}^{1/2}$

Let the $G.P.$ be $x,xy,x{y}^2,...$

Then $t_{p+q}=x{y}^{p+q-1}=a$ ...(1)

and $t_{p-q}=x{y}^{p-q-1}=b$ ...(2)

Dividing (1) by (2), $y^{2q}=\frac{a}{b}\Rightarrow y={\left(\frac{a}{b}\right)}^{1/2q}$

From (1), $x=a{\left(\frac{b}{a}\right)}^{(p+q-1)/\left(2q\right)}$

$\therefore t_p=x{y}^{p-1}=a.{\left(\frac{b}{a}\right)}^{(p+q-1)/\left(2q\right)}.{\left(\frac{a}{b}\right)}^{(p-1)/2q}$

$=a^{(1-\frac{p+q-1}{2q}+\frac{p-1}{2q})}.b^{(\frac{p+q-1}{2q}-\frac{p-1}{2q})}=a^{\frac{1}{2}}{b}^{\frac{1}{2}}=\sqrt{ab}$