Question

In a G.P. if the (m + n)^th term is p and (m − n)^th term is q, then its m^th term is 

विकल्प

• (a) 0 

• (b) pq

• (c) \[\sqrt{pq}\]

• (d) \[\frac{1}{2}(p + q)\] 

Answer 1

(c) \[\sqrt{pq}\] 

\[\text{ Here }, a_\left( m + n \right) = p\]
\[ \Rightarrow a r^\left( m + n - 1 \right) = p . . . . . . . (i)\]
\[\text{ Also }, a_\left( m - n \right) = q\]
\[ \Rightarrow a r^\left( m - n - 1 \right) = q . . . . . . . (ii)\]
\[\text{ Mutliplying } (i) \text{ and } (ii): \]
\[ \Rightarrow a r^\left( m + n - 1 \right) a r^\left( m - n - 1 \right) = pq\]
\[ \Rightarrow a^2 r^\left( 2m - 2 \right) = pq\]
\[ \Rightarrow \left( a r^\left( m - 1 \right) \right)^2 = pq\]
\[ \Rightarrow a r^\left( m - 1 \right) = \sqrt{pq}\]
\[ \Rightarrow a_m = \sqrt{pq}\]
\[\text{ Thus, the } m^{th} \text{ term is }  \sqrt{pq} . \]