  We first show the existence and nature of convergence to a limiting set of
roots for polynomials in a three-term recurrence of the form $p_{n+1}(z) =
Q_k(z)p_{n}(z)+ \gamma p_{n-1}(z)$ as $n$ $\rightarrow$ $\infty$, where the
coefficient $Q_k(z)$ is a $k^{th}$ degree polynomial, and $z,\gamma \in
\mathbb{C}$. We extend these results to relations for numerically approximating
roots of such polynomials for any given $n$. General solutions for the
evaluation are motivated by large computational efforts and errors in the
iterative numerical methods. Later, we apply this solution to the eigenvalue
problems represented by tridiagonal matrices with a periodicity $k$ in its
entries, providing a more accurate numerical method for evaluation of spectra
of chains and a reduction in computational effort from $\mathcal{O}(n^2)$ to
$\mathcal{O}(n)$. We also show that these results along with the spectral rules
of Kronecker products allow an efficient and accurate evaluation of spectra of
many spatial lattices and other periodic graphs.
