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} = ar^{4} = \frac{10}{9}\left(\frac{3}{5}\right)^{2} $

$ = \frac{10}{9} \times \frac{3\times3}{5\times5} = \frac{2}{5}$