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Question

Question: \[1 - \frac{1}{8} + \frac{1}{8}.\frac{3}{16} - ......... =\]...

1−18+18.316−.........=1 - \frac{1}{8} + \frac{1}{8}.\frac{3}{16} - ......... =

A

25\frac{2}{5}

B

25\frac{\sqrt{2}}{5}

C

25\frac{2}{\sqrt{5}}

D

None of these

Answer

25\frac{2}{\sqrt{5}}

Explanation

Solution

We know that (1+x)n=1+nx+n(n−1)2!x2+......∞(1 + x)^{n} = 1 + nx + \frac{n(n - 1)}{2!}x^{2} + ......\infty

Here nx=−18,n(n−1)2x2=38.16nx = - \frac{1}{8},\frac{n(n - 1)}{2}x^{2} = \frac{3}{8.16}⇒ x=14,n=−12x = \frac{1}{4},n = - \frac{1}{2}

⇒ (1+14)−1/2=25\left( 1 + \frac{1}{4} \right)^{- 1/2} = \frac{2}{\sqrt{5}}.