Question

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

• A. $5$
• B. $12$
• C. $10$
• D. $9$

Answer 1

Answer: (A), (D)

$5$
$9$

Given : $r^{th},(r+1{)}^{th}$ and $(r+2{)}^{th}$ terms in the expansion of ${(1+x)}^{14}$ are in A.P.

$\Rightarrow {}^m{C}_{r-1},{}^m{C}_r,{}^m{C}_{r+1}$ are in A.P.

$\Rightarrow 2{}^m{C}_r={}^m{C}_{r-1}+{}^m{C}_{r+1}$

$\Rightarrow 2=\frac{{}^m{C}_{r-1}+{}^m{C}_{r+1}}{{}^m{C}_r}$

$\Rightarrow \frac{r}{m-r+1}+\frac{m-r}{r+1}\Rightarrow m^2-m(4r+1)+4r^2-2=0$

Substitute $m=14$ ...... [Given]

$\Rightarrow r=5,9$

Hence, options A and D are correct.