First slide

Binomial theorem for positive integral Index

Question

If the coefficients of the $r t h, (r + 1) t h$ and $(r + 2) t h$ terms in the binomial expansion of ${(1 + y)}^{m}$ are in A.P., then m and r satisfy the equation

Moderate

A

$m^{2} - m (4 r + 1) + 4 r^{2} - 2 = 0$
B

$m^{2} - m (4 r - 1) + 4 r^{2} + 2 = 0$
C

$m^{2} - m (4 r - 1) + 4 r^{2} - 2 = 0$
D

$m^{2} - m (4 r + 1) + 4 r^{2} + 2 = 0$

Solution

Coefficient of the r th term is $^{m} C_{r - 1}$ According to the given condition $^{m} C_{r - 1} +^{m} C_{r + 1} = 2 (^{m} C_{r})$
$\begin{array}{ll} \Rightarrow \frac{^{m} C_{r - 1}}{^{m} C_{r}} + \frac{^{m} C_{r + 1}}{^{m} C_{r}} = 2 \Rightarrow \frac{r}{m + 1 - r} + \frac{m - r}{r + 1} = 2 \\ \Rightarrow r^{2} + r + (m - r) (m + 1 - r) = 2 (m + 1 - r) (r + 1) \\ \Rightarrow r^{2} + r + m^{2} + (1 - 2 r) m - r (1 - r) = \\ 2 m (r + 1) + 2 (1 - r^{2}) \\ \Rightarrow m^{2} - m (4 r + 1) + 4 r^{2} - 2 = 0 . \end{array}$

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