Infinite Sequences and Series · Hard problem

For a constant $p>0$, let $a_n=\dfrac{n+\sqrt n}{n^{p}-1}$. There is exactly one value of $p$ for which $\displaystyle\lim_{n\to\infty}\dfrac{a_n}{n^{-2}}$ is a finite nonzero number. Find that value of $p$.