Let $n$ be a fixed odd positive integer. For each odd prime $p$, define $$a_p=\frac{1}{p-1}\sum_{k=1}^{\frac{p-1}{2}}\bigg\{\frac{k^{2n}}{p}\bigg\}.$$Prove that there is a real number $c$ such that $a_p = c$ for infinitely many primes $p$. Note: $\left\{x\right\} = x - \left\lfloor x\right\rfloor$ is the fractional part of $x$.