The common ratio of 2 + 2/3 + 2/9 + 2/27 + …………….. isA) 1/2B

1.	The common ratio of 2 + 2/3 + 2/9 + 2/27 + …………….. isA) 1/2B) 2C) 3D) 1/3
Answer» Correct option is (D) 1/3 Let \(2,\frac{2}{3},\frac{2}{9},\frac{2}{27},.....\) be given sequence. \(a_1=2,a_2=\frac23,a_3=\frac29,a_4=\frac2{27}\) Now, \(\frac{a_2}{a_1}=\frac{\frac23}{2}=\frac{2}{6}=\frac{1}{3},\) \(\frac{a_3}{a_2}=\cfrac{\frac{2}{9}}{\frac{2}{3}}=\frac{2}{9}\times\frac{3}{2}=\frac{1}{3},\) \(\frac{a_4}{a_3}=\cfrac{\frac{2}{27}}{\frac{2}{9}}=\frac{2}{27}\times\frac{9}{2}=\frac{1}{3}\) \(\because\frac{a_4}{a_3}=\frac{a_3}{a_2}=\frac{a_2}{a_1}=\frac13\) \(\therefore\) Sequence \(2,\frac{2}{3},\frac{2}{9},\frac{2}{27},.....\) will form a G.P. whose common ratio is \(r=\frac{a_2}{a_1}=\frac13.\) Correct option is D) 1/3

The common ratio of 2 + 2/3 + 2/9 + 2/27 + …………….. isA) 1/2B) 2C) 3D) 1/3

Answer»

Correct option is (D) 1/3

Let \(2,\frac{2}{3},\frac{2}{9},\frac{2}{27},.....\) be given sequence.

\(a_1=2,a_2=\frac23,a_3=\frac29,a_4=\frac2{27}\)

Now, \(\frac{a_2}{a_1}=\frac{\frac23}{2}=\frac{2}{6}=\frac{1}{3},\)

\(\frac{a_3}{a_2}=\cfrac{\frac{2}{9}}{\frac{2}{3}}=\frac{2}{9}\times\frac{3}{2}=\frac{1}{3},\)

\(\frac{a_4}{a_3}=\cfrac{\frac{2}{27}}{\frac{2}{9}}=\frac{2}{27}\times\frac{9}{2}=\frac{1}{3}\)

\(\because\frac{a_4}{a_3}=\frac{a_3}{a_2}=\frac{a_2}{a_1}=\frac13\)

\(\therefore\) Sequence \(2,\frac{2}{3},\frac{2}{9},\frac{2}{27},.....\) will form a G.P. whose common ratio is \(r=\frac{a_2}{a_1}=\frac13.\)

Correct option is D) 1/3