<s>[INST] Solving Recurrence Relations using Iteration $$a_0 = 2; \qquad a_k=4a_{k-1}+5 ~ \forall\ k\ge 1$$
I have already tried solving for $a_1$ through $a_5$.
 [/INST] $$\begin{align*}
a_k - a_{k-1} & = 4a_{k-1}+5 -(4a_{k-2}+5) \\
&= 4(a_{k-1} - a_{k-2}) \\
&= 4^2(a_{k-2}-a_{k-3}) \\
& =4^{k-1}(a_1-a_0) \\
& =11\cdot 4^{k-1} \\
\Rightarrow a_k & = (a_k-a_{k-1})+(a_{k-1} - a_{k-2})+\cdots +(a_1-a_0)+a_0 \\
& = 11\cdot 4^{k-1} +11\cdot 4^{k-2} +\cdots + 11\cdot 4 + 11 + 2 \\
& = \ldots
\end{align*}$$
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