If the coefficients of 2nd, 3rd and 4th terms in the expansion of
Question:

If the coefficients of 2 nd, 3 rd and 4 th terms in the expansion of $(1+x)^{n}, n \in N$ are in A.P., then $n=$

(a) 7

(b) 14

(c) 2

(d) none of these

Solution:

(a) 7

Coefficients of the 2 nd, 3 rd and 4 th terms in the given expansion are:

${ }^{n} C_{1},{ }^{n} C_{2}$ and ${ }^{n} C_{3}$

We have :

$2 \times{ }^{n} C_{2}={ }^{n} C_{1}+{ }^{n} C_{3}$

Dividing both sides by ${ }^{n} C_{2}$, we get:

$2=\frac{{ }^{n} C_{1}}{{ }^{n} C_{2}}+\frac{{ }^{n} C_{3}}{{ }^{n} C_{2}}$

$\Rightarrow 2=\frac{2}{n-1}+\frac{n-2}{3}$

$\Rightarrow 6 n-6=6+n^{2}+2-3 n$

$\Rightarrow n^{2}-9 n+14=0$

$\Rightarrow n=7(\because n \neq 2$ as $2>3$ in the 4 th term $)$