Question

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

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

Answer 1

Answer: (C)

The given series is $\frac{10}{9},\frac{1}{3}\sqrt{\frac{20}{3}},\frac{2}{3},...$

Here, $a=\frac{10}{9}$

and $r=\frac{1}{3}\sqrt{\frac{20}{3}}\times \frac{9}{10}$

$=\sqrt{\frac{20\times 9\times 9}{9\times 3\times 100}}=\sqrt{\frac{3}{5}}$

$\therefore T_5=a{r}^4=\frac{10}{9}{\left(\frac{3}{5}\right)}^2$

$=\frac{10}{9}\times \frac{3\times 3}{5\times 5}=\frac{2}{5}$