In a G.P. ap + q= m,ap - q = n then ap = …………………A) mnB)�

1.	In a G.P. ap + q= m,ap - q = n then ap = …………………A) mnB) \(\sqrt{mn}\)C) \(\cfrac{mn}2\)D) \(\cfrac{m}2\)
Answer» Correct option is (B) \(\sqrt{mn} \) Let first term & common ratio of the G.P. is a & r respectively. Given that \(a_{P+Q}=m,a_{P-Q}=n\) \(\Rightarrow\) \(ar^{P+Q-1} = m\) _____________(1) and \(ar^{P-Q-1}=n\) _____________(2) \((\because a_n=ar^{n-1})\) Divide equation (1) by (2), we get \(\frac{ar^{P+Q-1}}{ar^{P-Q-1}}=\frac mn\) \(\Rightarrow r^{P+Q-1-(P-Q-1)}=\frac mn\) \(\Rightarrow r^{2Q}=\frac mn\) \(\Rightarrow r^Q=\sqrt{\frac mn}\) _____________(3) From (1), we have \(ar^{P+Q-1}=m\) \(\Rightarrow ar^{P-1}.r^Q=m\) \((\because a^{m+n}=a^m.a^n)\) \(\Rightarrow a_P.\sqrt{\frac mn}=m\) (From (3) \(\&\;a_P=ar^{P-1})\) \(\Rightarrow a_P=m.\sqrt{\frac nm}=\sqrt{mn}\) Correct option is B) \(\sqrt{mn}\)

In a G.P. ap + q= m,ap - q = n then ap = …………………A) mnB) \(\sqrt{mn}\)C) \(\cfrac{mn}2\)D) \(\cfrac{m}2\)

Answer»

Correct option is (B) \(\sqrt{mn} \)

Let first term & common ratio of the G.P. is a & r respectively.

Given that \(a_{P+Q}=m,a_{P-Q}=n\)

\(\Rightarrow\) \(ar^{P+Q-1} = m\) _____________(1)

and \(ar^{P-Q-1}=n\) _____________(2) \((\because a_n=ar^{n-1})\)

Divide equation (1) by (2), we get

\(\frac{ar^{P+Q-1}}{ar^{P-Q-1}}=\frac mn\)

\(\Rightarrow r^{P+Q-1-(P-Q-1)}=\frac mn\)

\(\Rightarrow r^{2Q}=\frac mn\)

\(\Rightarrow r^Q=\sqrt{\frac mn}\) _____________(3)

From (1), we have

\(ar^{P+Q-1}=m\)

\(\Rightarrow ar^{P-1}.r^Q=m\) \((\because a^{m+n}=a^m.a^n)\)

\(\Rightarrow a_P.\sqrt{\frac mn}=m\) (From (3) \(\&\;a_P=ar^{P-1})\)

\(\Rightarrow a_P=m.\sqrt{\frac nm}=\sqrt{mn}\)

Correct option is B) \(\sqrt{mn}\)