A) \[\frac{1}{\sqrt{5}}\]
B) \[\frac{1}{\sqrt{2}}\]
C) \[\sqrt{3}\]
D) \[\sqrt{\frac{5}{3}}\]
Correct Answer: D
Solution :
Let \[S=1+\frac{1}{5}+\frac{1.3}{5.10}+\frac{1.3.5}{5.10.15}+....\] We know that \[{{(1+x)}^{n}}=1\frac{nx}{1!}+\frac{n(n-1)}{2!}{{x}^{2}}\] \[+\frac{n(n-1)(n-2)}{3!}{{x}^{3}}+...\] \[\Rightarrow \]\[nx=\frac{1}{5}\]and \[\frac{n(n-1){{x}^{2}}}{2!}=\frac{1.3}{5.10}\] \[\Rightarrow \]\[n=-\frac{1}{2}\]and \[x=-\frac{2}{5}\] \[\therefore \] \[S={{\left( 1-\frac{2}{5} \right)}^{-1/2}}={{\left( \frac{3}{5} \right)}^{-1/2}}=\sqrt{\frac{5}{3}}\]
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