Consider $$\left(x^3+\frac2{x^2}\right)^n, \qquad x\ne0,$$ where $n$ is a positive integer. S

Question

Consider $$\left(x^3+\frac2{x^2}\right)^n, \qquad x\ne0,$$ where $n$ is a positive integer. Show that a constant term exists if and only if $n$ is a multiple of $5$. Find the constant term for the smallest possible positive value of $n$. For that value of $n$, explain why there is no term containing $x$.

Accepted Answer

(a) $5\mid n$; (b) $80$ when $n=5$; (c) required $r=\frac{14}{5}$ is invalid.

Binomial Theorem

Binomial Theorem