Mathematics - Binomial Theorem Question with Solution | TestHub

Let $r^{\text{th }}$ and $(r+1{)}^{\text{th }}$ term has equal coefficient $$\begin{array}{rl}& {\left(2+\frac{x}{3}\right)}^{55}=2^{55}{\left(1+\frac{x}{6}\right)}^{55}\\ & r^{\text{th }}\text{ term }=2^{5555}{C}_r{\left(\frac{x}{6}\right)}^r\end{array}$$ Coefficient of $x^r$ is $2^{55}{}^{55}{C}_r\frac{1}{6^r}$ $$(r+1{)}^{\text{th }}\text{ term }=2^{5555}{C}_{r+1}{\left(\frac{x}{6}\right)}^{r+1}$$ Coefficient of $x^{r+1}$ is $2^{55}{}^{55}{C}_{r+1}\cdot \frac{1}{6^{r+1}}$ Both coefficients are equal $$2^{5555}{C}_r\frac{1}{6^r}=2^{5555}{C}_{r+1}\frac{1}{6^{r+1}}$$ $$\begin{array}{rl}& \frac{1}{|r|55-r}=\frac{1}{\lfloor r+1\mid 54-r}\cdot \frac{1}{6}\\ & 6(r+1)=55-r\\ & 6r+6=55-r\\ & 7r=49\\ & r=7\\ & (r+1)=8\end{array}$$ Coefficient of $7^{\text{th }}$ and $8^{\text{th }}$ terms are equal.