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 :

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

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

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

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

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

$\Rightarrow r=7$

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