Question

If, in a G.P., the $(p+q)$ th term is $a$ and the $(p-q)$ th term is $b$, then $p$ th term is

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

Answer 1

Answer: (B)

Let the G.P. be $x,xy,x{y}^2,\dots$
Then $t_{p+q}=x{y}^{p+q-1}=a(1)$
and $t_{p-q}=x{y}^{p-q-1}=b(2)$
Dividing Eqs (1) by (2),
$y^{2q}=\frac{a}{b}$
$\therefore y={\left(\frac{a}{b}\right)}^{\frac{1}{2q}}$
From Eq. (1),
$x=a{\left(\frac{b}{a}\right)}^{\frac{p+q-1}{2q}}$
$\therefore t_p=x{y}^{p-1}=a\cdot {\left(\frac{b}{a}\right)}^{\frac{p+q-1}{2q}}{\left(\frac{a}{b}\right)}^{\frac{p-1}{2q}}$
$=1-\frac{p+q-1}{2q}+\frac{p-1}{2q}\times b^{\frac{p+q-1}{2q}-\frac{p-1}{2q}}$
$=a^{\frac{1}{2}}{b}^{\frac{1}{2}}=\sqrt{ab}$