First slide

Binomial theorem for positive integral Index

Question

In the expansion of $(1 + p x)^{n}, n \in N,$ the coefficient of $x$ and $x^{2}$ are $8$ and $24$ respectively, then

Moderate

A

$n = 3, p = 2$
B

$n = 4, p = 2$
C

$n = 4, p = 3$
D

$n = 5, p = 3$

Solution

We have
$\begin{aligned} (1 + p x)^{n} = 1 +^{n} C_{1} (p x) +^{n} C_{2} (p x)^{2} + \dots \\ = 1 + n p x + \frac{1}{2} n (n - 1) p^{2} x^{2} + \dots \end{aligned}$
According to the hypothesis,
$n p = 8$ and $\frac{1}{2} n (n - 1) p^{2} = 24$
Putting $p = 8 / n$ in the second expression, we get
$\begin{array}{l} \frac{1}{2} n (n - 1) {(\frac{8}{n})}^{2} = 24 \Rightarrow \frac{n - 1}{n} = \frac{24 \times 2}{8 \times 8} = \frac{3}{4} \\ \Rightarrow 4 n - 4 = 3 n \Rightarrow n = 4 \end{array}$
Putting this value in $n p = 8$ , we get $p = 2 .$

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