Let $S=\{1,5,9\}$. Prove that if $n\in S$ and $\frac{n^2+n-6}{2}$ is odd, then $\frac{2n^3+3n^2+n}{6}$ is even.

I first assumed $\frac{n^2+n-6}{2}$ is odd. Since $n\in S$, n must be either $\{1,5,9\}$ so I attempted to plug in all values into the fraction and I get even integers only.