Question

If the coeff of $2^{nd},3^{rd}$ and the $4^{th}$ terms in the expansion of $(1+x{)}^n$ are in A.P, then the value of $n$ is :

• A. $2$
• B. $7$
• C. $11$
• D. $14$

Answer 1

Answer: (B)

$7$

Solution:

We know that in the coefficient of $r^{th}$ term in expansion of $(1+x{)}^n={}^n{C}_{r-1}$

Now, Coefficient of $2^{nd}$ term in the expansion $(1+x{)}^n={}^n{C}_1$

Coefficient of $3^{rd}$ term in the expansion $(1+x{)}^n={}^n{C}_2$

Coefficient of $4^{th}$ term in the expansion $(1+x{)}^n={}^n{C}_3$

Since, ${}^n{C}_1,{}^n{C}_2$ and ${}^n{C}_3$ are in A.P

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

or, $\frac{n!}{1!(n-1)!}+\frac{n!}{3!(n-3)!}=2\times \frac{n!}{2!(n-2)!}$

or, $\frac{1}{(n-1)(n-2)}+\frac{1}{6}=\frac{1}{(n-2)}$

or, $6+(n-1)(n-2)=6(n-1)$

or, $6+n^2-3n+2=6n-6$

or, $n^2-9n+14=0$

or, $(n-7)(n-2)=0$

or, $n=7$ or $n=2$

If we take $n=2$ then number of terms in the expansion are $3$ which can't satisfy the above given condition. Therefore $n=7$ is the solution.

Hence, B is the correct option.