First slide

Geometric progression

Question

Suppose $(m + n)$ th term of a $G . P .$ is $p$ and $(m - n)$ th term is $q,$ then its $n$ th is

Moderate

A

$\sqrt{p q}$
B

$p {(\frac{q}{p})}^{m / n}$
C

$p {(\frac{q}{p})}^{m / 2 n}$
D

$\sqrt{\frac{p}{q}}$

Solution

Let a be the first term and r be the common ratio.
We have
$\begin{array}{l} p = a_{m + n} = a r^{m + n - 1} \\ q = a_{m - n} = a r^{m - n - 1} \\ \Rightarrow \frac{p}{q} = \frac{a_{m + n}}{a_{m - n}} = r^{2 n} \Rightarrow r = {(\frac{p}{q})}^{1 / 2 n} \end{array}$
Now, $a_{n} = a r^{n - 1} = \frac{a r^{m + n - 1}}{r^{m}}$
$= p {(\frac{p}{q})}^{- m / 2 n} = p {(\frac{q}{p})}^{m / 2 n}$

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