If the coefficients of $x^{7}$ and $x^{8}$ in the expansion of $\left(2+\frac{x}{3} ight)^{n}$\ ar

Question

If the coefficients of $x^{7}$ and $x^{8}$ in the expansion of $\left(2+\frac{x}{3}ight)^{n}$\ are equal, then the value of $n$ is equal to

Professor PC Thomas Classes · Answer

(55) : Coefficients of $x^{7}$ and $x^{8}$ in the expansion of $\left(2+\frac{x}{3}ight)^{n}$ are equal.\ $\Rightarrow{ }^{n} C_{7}(2)^{n-7}\left(\frac{1}{3}ight)^{7}={ }^{n} C_{8}(2)^{n-8}\left(\frac{1}{3}ight)^{8}$\ $\Rightarrow \frac{{ }^{n} C_{8}}{{ }^{n} C_{7}}=3 \cdot 2=6 \Rightarrow \frac{n-7}{8}=6$\ $\Rightarrow n=55$