Question

If $(2+\frac{x}{3}{)}^{55}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are :

• A. $7^{th}$ and $8^{th}$
• B. $8^{th}$ and $9^{th}$
• C. ${28}^{th}$ and ${29}^{th}$
• D. ${27}^{th}$ and ${28}^{th}$

Answer 1

Answer: (B)

$8^{th}$ and $9^{th}$

Coefficient of $x^r$ and $x^{r+1}$ are equal.

Coefficient of $T_{r+1}=$ Coefficient of $T_{r+2}$

$\Rightarrow {{}^{55}C}_r{\left(2\right)}^{55-r}{\left(\frac{1}{3}\right)}^r={{}^{55}C}_{r+1}{\left(2\right)}^{54-r}{\left(\frac{1}{3}\right)}^{r+1}$

.
$\frac{55!}{(55-r)!r!}\left(2\right)=\frac{55!}{(54-r)!(r+1)!}\frac{1}{3}$

$\Rightarrow 6(r+1)=55-r$

$\Rightarrow r=7$

So, the terms will be $8^{th}$ and $9^{th}$