Question

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

• A. $5$
• B. $6$
• C. $7$
• D. $9$

Answer 1

Answer: (A), (D)

$5$
$9$

Since, coefficients of $r^{th},(r+1{)}^{th}$ and $(r+2{)}^{th}$ terms in $(1+x{)}^{14}$ are in A.P.
$\Rightarrow 2{(}^{14}{C}_r){=}^{14}{C}_{r-1}{+}^{14}{C}_{r+1}$
$\frac{{}^{14}{C}_{r-1}}{{}^{14}{C}_r}+\frac{{}^{14}{C}_{r+1}}{{}^{14}{C}_r}=2$
$\frac{r}{14-r+1}+\frac{14-r}{r+1}=2$
$\Rightarrow \frac{r^2+r+(15-r)(14-r)}{(15-r)(r+1)}=2$
$\Rightarrow 2r^2-18r+210+2(15-r)(r+1)$
$\Rightarrow 4r^2-56r+180=0$
$\Rightarrow r^2-14r+45=0$
$\Rightarrow r=5,9$