Let $\mathrm{D}_{\mathrm{k}}=\left|\begin{array}{ccc}1 & 2 k & 2 k-1 \\ n & n^{2}+n+2 & n^{2} \\ n & n^{2}+n & n^{2}+n+2\end{array}\right|$. If $\sum_\limits{k=1}^{n} \mathrm{D}_{\mathrm{k}}=96$, then $n$ is equal to _____________.