First slide

Binomial theorem for positive integral Index

Question

The sum of the series $S = \frac{1}{19!} + \frac{1}{3! 17!} + \frac{1}{5! 15!} + \dots \dots \dots t o 10$ terms is equal to

Moderate

A

$\frac{2^{19}}{20!}$
B

$\frac{2^{20}}{20!}$
C

$\frac{2^{10}}{20!}$
D

$\frac{2^{19}}{19!}$

Solution

We have,
$\begin{array}{l} S = \frac{1}{19!} + \frac{1}{3! 17!} + \frac{1}{5! 15!} + \dots \dots \dots to 10 terms \\ S = \frac{1}{20!} \{\frac{20!}{1! 19!} + \frac{20!}{3! 17!} + \frac{20!}{5! 15!} + \dots \dots \dots . t \circ 10 terms\} \\ S = \frac{1}{20!} \{^{20} C_{1} +^{20} C_{3} +^{20} C_{5} + \dots \dots \dots +^{20} C_{19}\} \\ S = \frac{1}{20!} \{2^{20 - 1}\} = \frac{2^{19}}{20!} \end{array}$

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