If the coefficient of $x^7$ and $x^8$ in the expansion of $\left(2+\frac{x}{3}\right)^n$ are equal,

Question

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

Professor PC Thomas Classes · Answer

Coeff of $x^7$ is ${ }^nC_7 2^{n-7} (1/3)^7$. Coeff of $x^8$ is ${ }^nC_8 2^{n-8} (1/3)^8$. Equating gives $n = 55$.