First slide

Geometric progression

Question

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

Moderate

A

$(ab)^{1 / 3}$
B

$(ab)^{1 / 2}$
C

$(ab)^{1 / 4}$
D

None of these

Solution

Let the G.P be $x, xy, {xy}^{2}, \dots$
then $t_{p + q} = {xy}^{p + q - 1} = a$ (1)
and $t_{p - q} = {xy}^{p - q - 1} = b$ (2)
Dividing Eqs (1) by (2),
$y^{2 q} = \frac{a}{b}$
$∴ y = {(\frac{a}{b})}^{\frac{1}{2 q}}$
From Eq. (1),
$x = a {(\frac{b}{a})}^{\frac{p + q - 1}{2 q}}$
$\begin{array}{l} ∴ t_{p} = {xy}^{p - 1} = a \cdot {(\frac{b}{a})}^{\frac{p + q - 1}{2 q}} {(\frac{a}{b})}^{\frac{p - 1}{2 q}} \\ = 1 - \frac{p + q - 1}{2 q} + \frac{p - 1}{2 q} \times b^{\frac{p + q - 1}{2 q} - \frac{p - 1}{2 q}} \\ = a^{\frac{1}{2}} b^{\frac{1}{2}} = \sqrt{ab} \end{array}$

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