Question

The $5^{th}$ term of the sequence $\frac{10}{9},\frac{1}{3},\frac{\sqrt{20}}{3},\frac{2}{3},\cdots$ is

• A. $\frac{1}{3}$
• B. $1$
• C. $2$
• D. $\frac{\sqrt{5}}{3}$

Answer 1

Answer: (C)

$2$

The series is divided in odd and even terms. The first series is $\frac{10}{9},\frac{\sqrt{20}}{3},...$ and the second series is $\frac{1}{3},\frac{2}{3},...$

The $5^{th}$ term will be a part of the $1^{st}$ series.

The general term of the series is $t_{2r+1}=\frac{3}{\sqrt{5}}\times t_{2r-1}$

$t_3=\frac{3}{\sqrt{5}}\times t_1=\frac{10}{9}\times \frac{3}{\sqrt{5}}=\frac{\sqrt{20}}{3}$

$t_5=\frac{3}{\sqrt{5}}\times t_3=\frac{3}{\sqrt{5}}\times \frac{\sqrt{20}}{3}=2$

Hence, option $C$ is correct.