If the $(p + q)$ th term of a geometric series is $m$ and the $(p - q)$ th term is $n$ , then the $p$ th term is

${(m n)}^{\frac{1}{2}}$
$m n$
$m + n$
$m - n$

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Solution

The correct option is A
${(m n)}^{\frac{1}{2}}$

Step1: Use the formula of the general term of a GP
Let $a$ be the first term and $r$ be the common ratio.
For a GP, the nth term $T_{n} = a r^{n - 1}$
$(p + q) t h t e r m = a r^{p + q - 1} = m . . \dots (i)$
$(p - q) t h t e r m = a r^{p - q - 1} = n . . \dots (i i)$
Multiply (i) and (ii)
$a r^{p + q - 1} \cdot a r^{p - q - 1} = m n$
Step 2: Simplify the above equation
$a^{2} r^{2 p - 2} = m n$
${(a r^{p - 1})}^{2} = m n$
$(a r^{p - 1}) = \sqrt (m n) = {(m n)}^{\frac{1}{2}}$
$p t h$ term is given by $a r^{p - 1}$
$a r^{p - 1} = {(m n)}^{\frac{1}{2}}$
$a_{p} = {(m n)}^{\frac{1}{2}}$
Therefore, the $p t h$ term is ${(m n)}^{\frac{1}{2}}$ .
Hence option A is the answer.

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