Question

If the coefficients of $r^{\text{th}},(r+1{)}^{\text{th}}\text{and}(r+2{)}^{\text{th}}$ terms in the expansion of $(1+x{)}^{14}$ are in A.P., then value $r$ can be

• A. $5$
• B. $12$
• C. $10$
• D. $9$

Answer 1

Answer: (A), (D)

$9$

Coefficients of $r^{\text{th}},(r+1{)}^{\text{th}}\text{and}(r+2{)}^{\text{th}}$ terms are ${}^{14}{C}_{r-1},{}^{14}{C}_r\text{and}{}^{14}{C}_{r+1}$
If these coefficients are in A.P., then
$2{(}^{14}{C}_r)={}^{14}{C}_{r-1}+{}^{14}{C}_{r+1}$
$\Rightarrow \frac{2\left(14\right)!}{r!(14-r)!}=\frac{\left(14\right)!}{(r-1)!(15-r)!}+\frac{\left(14\right)!}{(r+1)!(13-r)!}$
$\Rightarrow \frac{2\left(14\right)!}{r!(14-r)!}=\frac{\left(14\right)!\left[\right(r+1)r+(15-r\left)\right(14-r)}{(r+1)!(15-r)!}$
$\Rightarrow 2(15-r)(r+1)=2r^2-28r+210$
$\Rightarrow r^2-14r+45=0$
$\Rightarrow (r-5)(r-9)=0$
$\Rightarrow r=5\text{or}9$