Diagonalization & Matrix Powers · Hard problem

A population of juveniles $j_k$ and adults $a_k$ evolves by $j_{k+1} = j_k + 4a_k$ and $a_{k+1} = \tfrac12 j_k$. It starts with $j_0 = 10$ and $a_0 = 10$. The number of juveniles grows roughly like $c\cdot 2^k$. Find $c = \displaystyle\lim_{k\to\infty}\dfrac{j_k}{2^k}$.