Let $A$ be a $2 \times 2$ matrix with $\det(A) = 1$ and $\text{tr}(A) = 2.5$. If $A^n = \begin{pmatrix} a_n & b_n \\ c_n & d_n \end{pmatrix}$, what is the limit of $\frac{a_n}{\text{tr}(A^n)}$ as $n \to \infty$?