First slide

Binomial theorem for positive integral Index

Question

Sum of the series $\frac{1}{1 n - 1} + \frac{1}{3 n - 3} + \frac{1}{5 n - 5} + ..... + \frac{1}{n - 1 1}$ is

Moderate

A

$\frac{2^{n - 1}}{n - 1}$
B

$\frac{2^{n - 1}}{n}$
C

$\frac{2^{n}}{n - 1}$
D

$\frac{2^{n}}{n}$

Solution

$S_{n} = \frac{1}{n} [\frac{n}{1} + \frac{n (n - 1) (n - 2)}{3} + \frac{n (n - 1) (n - 2) (n - 3) (n - 4)}{5} + .....]$
$= \frac{1}{n} [^{n} C_{1} +^{n} C_{3} +^{n} C_{5} + ....] = \frac{1}{n} [C_{1} + C_{3} + C_{5} + ....] = \frac{1}{n} 2^{n - 1}$ .

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