Question

$1+\frac{2}{3}.\frac{1}{2}+\frac{2.5}{3.6}{\left(\frac{1}{2}\right)}^2.+\frac{\mathrm{2.5.8}}{\mathrm{3.6.9}}{\left(\frac{1}{2}\right)}^3$ + .... =

• A. 2^1/3
• B. 3^1/4
• C. 4^1/3
• D. 3^1/3

Answer 1

The correct option is C

4^1/3

$S=1+\frac{2}{3}.\frac{1}{2}+\frac{2.5}{3.6}{\left(\frac{1}{2}\right)}^2+\frac{\mathrm{2.5.8}}{\mathrm{3.6.9}}{\left(\frac{1}{2}\right)}^3+....=1+\frac{\frac{2}{3}}{1}\left(\frac{1}{2}\right)+\frac{\left(\frac{2}{3}\right)\left(\frac{5}{3}\right)}{2!}{\left(\frac{1}{2}\right)}^2+\frac{\left(\frac{2}{3}\right)\left(\frac{5}{3}\right)\left(\frac{8}{3}\right)}{3!}{\left(\frac{1}{2}\right)}^3+......={(1-\frac{1}{2})}^{-\frac{2}{3}}=2^{\frac{2}{3}}=4^{\frac{1}{3}}$