Continued Matrices

Algebra Level 4

If \[\begin{bmatrix} 1 & 1\\ 1 & 0 \end{bmatrix} ^ k = \begin{bmatrix} a_{k} & b_{k}\\ c_{k} & d_{k} \end{bmatrix},\] find the closed form of \(\displaystyle \lim_{k\to \infty} \frac{a_{k}}{c_{k}}\) to 3 decimal places.

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