First slide

Binomial theorem for positive integral Index

Question

The sum of the series $\sum_{r = 0}^{10}^{20} C_{r'}$ is

Moderate

A

$2^{20}$
B

$2^{19}$
C

$2^{19} + {\frac{1}{2}}^{20} C_{10}$
D

$2^{19} - {\frac{1}{2}}^{20} C_{10}$

Solution

$\sum_{r = 0}^{10}^{20} C_{r}$
$\begin{array}{l} (^{20} C_{0} +^{20} C_{1} + \dots +^{20} C_{10}) \\ = \frac{1}{2} \{2 (^{20} C_{0} +^{20} C_{1} + \dots +^{20} C_{10})\} \\ = \frac{2}{2} \{(^{20} C_{0} +^{20} C_{1} + \dots +^{20} C_{10}) + (^{20} C_{20} +^{20} C_{19} + \dots +^{20} C_{10})\} \\ = \frac{1}{2} \{(^{20} C_{0} +^{20} C_{1} + \dots +^{20} C_{10} +^{20} C_{11} + \dots +^{20} C_{20}) +^{20} C_{10}\} \end{array}$
$= \frac{1}{2} \{2^{20} +^{20} C_{10}\} = 2^{19} + {\frac{1}{2}}^{20} C_{10}$

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