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 value of $n$ is

• A. $2$
• B. $7$
• C. $11$
• D. $14$

Answer 1

Answer: (B)

Given expansion is $(1+x{)}^n$.
Coefficients of $2^{nd}$, $3^{rd}$ and $4^{th}$ terms are ${}^n{C}_1$, ${}^n{C}_2$ and ${}^n{C}_3$ respectively.
Since, ${}^n{C}_1$, ${}^n{C}_2$, ${}^n{C}_3$ are in A.P.
$\Rightarrow 2^n{C}_2={}^n{C}_1+{}^n{C}_3$
$\Rightarrow 2=\frac{{}^n{C}_1}{{}^n{C}_2}+\frac{{}^n{C}_3}{{}^n{C}_2}$
$\Rightarrow 2=\frac{2}{n-1}+\frac{n-2}{3}$
$\Rightarrow n^2-9n+14=0$
$\Rightarrow n=2$, $7$
But $n\ne 2$
$\therefore n=7$