Binomial Theorem Ques 25
Let $n$ be a positive integer. If the coefficients of $2 \mathrm{nd}, 3 \mathrm{rd}$, and 4th terms in the expansion of $(1+x)^{n}$ are in AP, then the value of $n$ is… .
(1994, 2M)
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Answer:
Correct Answer: 25.$(n=7)$
Solution:
Formula:
Properties of binomial theorem:
- Let the coefficients of $2$ nd, $3$ rd and $4$ th terms in the expansion of $(1+x)^{n}$ is ${ }^{n} C_{1},{ }^{n} C_{2},{ }^{n} C_{3}$.
According to given condition,
$ 2\left({ }^{n} C_{2}\right) ={ }^{n} C_{1}+{ }^{n} C_{3} $
$\Rightarrow 2 \frac{n(n-1)}{1 \cdot 2} =n+\frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3} $
$\Rightarrow n-1 =1+\frac{(n-1)(n-2)}{6} $
$\Rightarrow n-1 =1+\frac{n^{2}-3 n+2}{6} $
$\Rightarrow 6 n-6 =6+n^{2}-3 n+2 $
$\Rightarrow n^{2}-9 n+14 =0 $
$\Rightarrow (n-2)(n-7) =0 $
$\Rightarrow n =2 $
$\text { or } n =7$
But ${ }^{n} C_{3}$ is true for $n \geq 3$, therefore $n=7$ is the answer.
BETA
