1 - 1 8 + 1 8 .3 16 - 1.3.5 8.16.24 + .....= - Vidyadip

MCQ

$1 - \frac{1}{8} + \frac{1}{8}.\frac{3}{{16}} - \frac{{1.3.5}}{{8.16.24}} + .....$=

A
$\frac{2}{5}$
B
$\frac{{\sqrt 2 }}{5}$
✓
$\frac{2}{{\sqrt 5 }}$
D
None of these

✓

Answer

Correct option: C.

$\frac{2}{{\sqrt 5 }}$

c
(c) Comparing the given expression to

$1 + nx + \frac{{n(n - 1)}}{{2!}}{x^2} + ....i.e.{(1 + x)^n},$we get

$nx = - \frac{1}{8}$and $\frac{{n(n - 1)}}{{2!}}{x^2} = \frac{3}{{128}} \Rightarrow x = \frac{1}{4},n = - \frac{1}{2}$

Hence $1 - \frac{1}{8} + \frac{1}{8}.\frac{3}{{16}} - .... = {\left( {1 + \frac{1}{4}} \right)^{ - 1/2}} = \frac{2}{{\sqrt 5 }}$

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