First slide

Binomial theorem for positive integral Index

Question

Sum of the series $\sum_{r = 0}^{10} (- 1)^{r} (^{10} C_{r}) [\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}}]$ is

Moderate

A

$\frac{2^{10} + 1}{2^{20}}$
B

$\frac{2^{10} - 1}{2^{20}}$
C

$\frac{2^{20} - 1}{2^{20}}$
D

none of these

Solution

$\begin{array}{l} \sum_{r = 0}^{10} (- 1)^{r} (^{10} C_{r}) [\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}}] \\ = {(1 - \frac{1}{2})}^{10} + {(1 - \frac{3}{4})}^{10} = \frac{2^{10} + 1}{2^{20}} \end{array}$

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