First slide

Binomial theorem for positive integral Index

Question

The coefficient of $x^{k}$ in the expansion of $E = 1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n}$ is

Moderate

A

$^{n} C_{k}$
B

$^{n + 1} C_{k}$
C

$^{n + 1} C_{k + 1}$
D

none of these

Solution

We have
$\begin{array}{l} E = 1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n} \\ = \frac{1 - (1 + x)^{n + 1}}{1 - (1 + x)} = \frac{1}{x} [(1 + x)^{n + 1} - 1] \end{array}$
Thus, coefficient of $x^{k}$ in $E$
$=$ Coefficient of $x^{k + 1}$ in $[(1 + x)^{n + 1} - 1]$
$=^{n + 1} C_{k + 1}$

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