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

Answer 1

The correct option is A

7

Coefficient of the 2nd, 3rd and 4th 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:

$2=\frac{{}^n{C}_1}{{}^n{C}_2}+\frac{{}^n{C}_3}{{}^n{C}_2}$

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

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

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

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