Question 25
The sequence $\alpha_v$ satisfies the recurrence relation: $\alpha_1 = 1$ and $\alpha_v = 5\alpha_{v-1} + 3^{v-1}$, $v \ge 2$. Determine the general term $\alpha_v$ and the greatest power of $2$ which divides the term $\alpha_k$, where $k=2^{2019}$.