Question

the series $1+\frac{1}{5}+\frac{1 \cdot 3}{5 \cdot 10}+\frac{1 \cdot 3 \cdot 5}{5 \cdot 10 \cdot 15}+\ldots$ is equal to:

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

Answer 1

Answer: (D)

Let $S=1+\frac{1}{5}+\frac{1 \cdot 3}{5 \cdot 10}+\frac{1 \cdot 3 \cdot 5}{5 \cdot 10 \cdot 15}+\ldots .$
We know that
$(1+x)^{n}=1+\frac{n x}{1 !}+\frac{n(n-1)}{2 !} x^{2}$
$+\frac{n(n-1)(n-2)}{3 !} x^{3}+\ldots$
$\Rightarrow n x=\frac{1}{5}$ and $\frac{n(n-1) x^{2}}{2 !}$
$=\frac{1 \cdot 3}{5 \cdot 10}$
$\Rightarrow n=-\frac{1}{2}$ and $x=-\frac{2}{5}$
$\therefore S=\left(1-\frac{2}{5}\right)^{-1 / 2}$
$=\left(\frac{3}{5}\right)^{-1 / 2}=\sqrt{\frac{5}{3}}$