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Question
In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is
Options
(a) 0
(b) pq
(c) \[\sqrt{pq}\]
(d) \[\frac{1}{2}(p + q)\]
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Solution
(c) \[\sqrt{pq}\]
\[\text{ Here }, a_\left( m + n \right) = p\]
\[ \Rightarrow a r^\left( m + n - 1 \right) = p . . . . . . . (i)\]
\[\text{ Also }, a_\left( m - n \right) = q\]
\[ \Rightarrow a r^\left( m - n - 1 \right) = q . . . . . . . (ii)\]
\[\text{ Mutliplying } (i) \text{ and } (ii): \]
\[ \Rightarrow a r^\left( m + n - 1 \right) a r^\left( m - n - 1 \right) = pq\]
\[ \Rightarrow a^2 r^\left( 2m - 2 \right) = pq\]
\[ \Rightarrow \left( a r^\left( m - 1 \right) \right)^2 = pq\]
\[ \Rightarrow a r^\left( m - 1 \right) = \sqrt{pq}\]
\[ \Rightarrow a_m = \sqrt{pq}\]
\[\text{ Thus, the } m^{th} \text{ term is } \sqrt{pq} . \]
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