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Question
If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is \[\left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\].
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Solution
\[\text { As, } a_p = q\]
\[ \Rightarrow a r^\left( p - 1 \right) = q . . . . . \left( i \right)\]
\[\text { Also, } a_q = p\]
\[ \Rightarrow a r^\left( q - 1 \right) = p . . . . . \left( ii \right)\]
\[\text { Dividing }\left( i \right)\text { by } \left( ii \right), \text { we get }\]
\[\frac{a r^\left( p - 1 \right)}{a r^\left( q - 1 \right)} = \frac{q}{p}\]
\[ \Rightarrow r^\left( p - 1 - q + 1 \right) = \frac{q}{p}\]
\[ \Rightarrow r^\left( p - q \right) = \frac{q}{p}\]
\[ \Rightarrow r = \left( \frac{q}{p} \right)^\frac{1}{\left( p - q \right)} \]
\[\text { Substituting the value of r in } \left( ii \right), \text { we get }\]
\[a \left[ \left( \frac{q}{p} \right)^\frac{1}{\left( p - q \right)} \right]^\left( q - 1 \right) = p\]
\[ \Rightarrow a\left[ \left( \frac{q}{p} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \right] = p\]
\[ \Rightarrow a = p \times \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \]
\[ \Rightarrow a = p \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \]
\[\text { Now, } \]
\[ a_\left( p + q \right) = a r^\left( p + q - 1 \right) \]
\[ = p \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \times \left[ \left( \frac{q}{p} \right)^\frac{1}{\left( p - q \right)} \right]^\left( p + q - 1 \right) \]
\[ = p \left( \frac{p}{q} \right)^\frac{\left( q - 1 \right)}{\left( p - q \right)} \times \left( \frac{q}{p} \right)^\frac{\left( p + q - 1 \right)}{\left( p - q \right)} \]
\[ = p \left( \frac{q}{p} \right)^\frac{- \left( q - 1 \right)}{\left( p - q \right)} \times \left( \frac{q}{p} \right)^\frac{\left( p + q - 1 \right)}{\left( p - q \right)} \]
\[ = p \times \left( \frac{q}{p} \right)^\frac{- \left( q - 1 \right)}{\left( p - q \right)} + \frac{\left( p + q - 1 \right)}{\left( p - q \right)} \]
\[ = p \times \left( \frac{q}{p} \right)^\frac{- q + 1 + p + q - 1}{\left( p - q \right)} \]
\[ = p \times \left( \frac{q}{p} \right)^\frac{p}{\left( p - q \right)} \]
\[ = \frac{p \times q^\frac{p}{\left( p - q \right)}}{p^\frac{p}{\left( p - q \right)}}\]
\[ = \frac{q^\frac{p}{\left( p - q \right)}}{p^\frac{p}{\left( p - q \right)} - 1}\]
\[ = \frac{q^\frac{p}{\left( p - q \right)}}{p^\frac{p - p + q}{\left( p - q \right)}}\]
\[ = \frac{q^\frac{p}{\left( p - q \right)}}{p^\frac{q}{\left( p - q \right)}}\]
\[ = \frac{q^{p \times \frac{1}{\left( p - q \right)}}}{p^{q \times \frac{1}{\left( p - q \right)}}}\]
\[ = \left( \frac{q^p}{p^q} \right)^\frac{1}{p - q}\]
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