Question

If coefficients of $5^{\text{th}},6^{\text{th}}$ and $7^{\text{th}}$ terms in the expansion of $(1+x{)}^n$ are in A.P., then the value(s) of $n$ is/are

• A. $7$
• B. $14$
• C. $8$
• D. $16$

Answer 1

Answer: (A), (B)

The correct options are
A $7$
B $14$

Given : $(1+x{)}^n$
General term is
$T_{r+1}={}^n{C}_r{1}^{n-r}{x}^r\Rightarrow T_{r+1}={}^n{C}_r{x}^r$
So, the coefficient are
$T_5={}^n{C}_4{T}_6={}^n{C}_5{T}_7={}^n{C}_6$
As $5^{\text{th}},6^{\text{th}}$ and $7^{\text{th}}$ terms are in A.P., so
$2{}^n{C}_5={}^n{C}_4{+}^n{C}_6\Rightarrow 2\left[\frac{n!}{(n-5)!5!}\right]=[\frac{n!}{(n-4)!4!}+\frac{n!}{(n-6)!6!}]\Rightarrow 2\left[\frac{1}{(n-5)5}\right]=[\frac{1}{(n-4)(n-5)}+\frac{1}{6\times 5}]\Rightarrow 12(n-4)=30+(n-5)(n-4)\Rightarrow 12n-48=n^2-9n+50\Rightarrow n^2-21n+98=0\Rightarrow (n-7)(n-21)=0\therefore n=7,14$