Infinite Sequences and Series · Medium problem

A term $a_n=\dfrac{cn+4}{n^{5/2}+1}$ shrinks at the same rate as $\dfrac1{n^{3/2}}$, in the sense that $\displaystyle\lim_{n\to\infty}\frac{a_n}{n^{-3/2}}$ is a finite nonzero number. If that limiting ratio equals $5$, find $c$.