First slide

Binomial theorem for positive integral Index

Question

The coefficient of $x^{m} in (1 + x)^{m} + (1 + x)^{m + 1} +, \dots, + (1 + x)^{n}, m \leq n$ is

Moderate

A

$^{n} C_{m}$
B

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

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

none of these

Solution

The coefficient of x_m in
$\begin{aligned} (1 + x)^{m} + (1 + x)^{m + 1} + (1 + x)^{m + 2} + \dots + (1 + x)^{n} \\ =^{m} C_{m} +^{m + 1} C_{m} +^{m + 2} C_{m} + \dots +^{n} C_{m} \\ =^{m + 1} C_{m + 1} +^{m + 1} C_{m} +^{m + 2} C_{m} + \dots +^{n} C_{m} \\ (∵^{m} C_{m} =^{m + 1} C_{m + 1} = 1) \\ =^{m + 2} C_{m + 1} +^{m + 2} C_{m} + \dots +^{n} C_{m} \\ (∵^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{r + 1}) \\ =^{m + 3} C_{m + 1} + \dots +^{n} C_{m} =^{n + 1} C_{m + 1} . \end{aligned}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App