First slide

Binomial theorem for positive integral Index

Question

The sum of the last 30coefficients of power sof x the binomial expansion of $(1 + x)^{59}$ is

Moderate

A

$2^{58}$
B

$2^{29}$
C

$2^{28}$
D

$2^{59} - 2^{29}$

Solution

We have,
$\begin{aligned} (1 + x)^{59} = \sum_{r = 0}^{59}^{59} C_{r} x^{r} \\ \Rightarrow 2^{59} = \sum_{r = 0}^{59}^{59} C_{r} \\ \Rightarrow 2^{59} = 2 (\sum_{r = 0}^{29}^{59} C_{r}) \\ \Rightarrow 2^{58} = \sum_{r = 0}^{29}^{59} C_{r} \\ \Rightarrow 2^{58} = \sum_{r = 0}^{29}^{59} C_{59} \dots \end{aligned}$
$\Rightarrow \sum_{r = 0}^{29}^{59} C_{59 - r} = 2^{58}$

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