Which term of the G.P. 1/3 , 1/9 , 1/27 , ..............is 1/(2187)A)

1.	Which term of the G.P. 1/3 , 1/9 , 1/27 , ..............is 1/(2187)A) 5thB) 6th C) 7th D) 8th
Answer» Correct option is (C) 7^th Given G.P. is \(\frac{1}{3},\frac{1}{9},\frac{1}{27},........\) \(\therefore a_1=\frac13,a_2=\frac19\) \(\therefore\) Common ratio = r \(=\frac{a_2}{a_1}=\cfrac{\frac19}{\frac13}\) \(=\frac19\times3=\frac13\) Let \(a_n\) = \(\frac{1}{2187}\) \(\Rightarrow\) \(ar^{n-1}\) = \(\frac{1}{2187}\) \((\because a_n=ar^{n-1}\) for GP) \(\Rightarrow\) \(r^{n-1}\) = \(\frac{1}{2187a}\) \(\Rightarrow\) \(r^{n-1}\) \(=\frac{3}{2187}=\frac1{729}\) \((\because a=\frac13)\) \(\Rightarrow(\frac13)^{n-1}=(\frac13)^6\) \((\because r=\frac13)\) \(\Rightarrow n-1=6\) (By comparing) \(\Rightarrow n=6+1=7\) Hence, \(\frac{1}{2187}\) is \(7^{th}\) term of given G.P. Correct option is C) 7^th

Which term of the G.P. 1/3 , 1/9 , 1/27 , ..............is 1/(2187)A) 5thB) 6th C) 7th D) 8th

Answer»

Correct option is (C) 7^th

Given G.P. is \(\frac{1}{3},\frac{1}{9},\frac{1}{27},........\)

\(\therefore a_1=\frac13,a_2=\frac19\)

\(\therefore\) Common ratio = r

\(=\frac{a_2}{a_1}=\cfrac{\frac19}{\frac13}\)

\(=\frac19\times3=\frac13\)

Let \(a_n\) = \(\frac{1}{2187}\)

\(\Rightarrow\) \(ar^{n-1}\) = \(\frac{1}{2187}\) \((\because a_n=ar^{n-1}\) for GP)

\(\Rightarrow\) \(r^{n-1}\) = \(\frac{1}{2187a}\)

\(\Rightarrow\) \(r^{n-1}\) \(=\frac{3}{2187}=\frac1{729}\) \((\because a=\frac13)\)

\(\Rightarrow(\frac13)^{n-1}=(\frac13)^6\) \((\because r=\frac13)\)

\(\Rightarrow n-1=6\) (By comparing)

\(\Rightarrow n=6+1=7\)

Hence, \(\frac{1}{2187}\) is \(7^{th}\) term of given G.P.

Correct option is C) 7^th