First slide

Binomial theorem for positive integral Index

Question

In the expansion of the following expression $1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n}$ the coefficient of $x^{k} (0 \leq k \leq n)$ is

Moderate

A

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

$^{n} C_{k}$
C

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

None of these

Solution

The given expression is
$1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n}$ being in GP.
Let
$\begin{aligned} S = 1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n} \\ = \frac{(1 + x)^{n + 1} - 1}{(1 + x) - 1} = x^{- 1} [(1 + x)^{n + 1} - 1] \end{aligned}$
$∴ The coefficient of x^{k} in S .$
$= The coefficient of x^{k + 1} in [(1 + x)^{n + 1} - 1] =^{n +} C_{k + 1}$

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