Question:
If the coefficients of 2 nd, 3 rd and 4 th terms in the expansion of $(1+x)^{n}$ are in A.P., then find the value of $n$.
Solution:
Coefficients of the $2 \mathrm{nd}, 3 \mathrm{rd}$ 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 $)$
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