Question

If the coefficients of 2nd, 3rd and 4th terms in the expansion of ${\left(1+x\right)}^n,n\in N$ are in A.P., then n =
(a) 7
(b) 14
(c) 2
(d) none of these

Answer 1

(a) 7
Coefficients of the 2nd, 3rd and 4th terms in the given expansion are:
$$\begin{gathered} {}^{n}C_1,{}^{n}C_2\mathrm{and}{}^{n}C_3 \\ \mathrm{We}\mathrm{have}: \\ 2\times {}^{n}C_2={}^{n}C_1+{}^{n}C_3 \\ \mathrm{Dividing}\mathrm{both}\mathrm{sides}\mathrm{by}{}^{n}C_2,\mathrm{we}\mathrm{get}: \\ 2=\frac{{}^{n}C_1}{{}^{n}C_2}+\frac{{}^{n}C_3}{{}^{n}C_2} \\ \Rightarrow 2=\frac{2}{n-1}+\frac{n-2}{3} \\ \Rightarrow 6n-6=6+n^2+2-3n \\ \Rightarrow n^2-9n+14=0 \\ \Rightarrow n=7\left(\because n\ne 2\mathrm{as}2>3\mathrm{in}\mathrm{the}4\mathrm{th}\mathrm{term}\right) \end{gathered}$$