Investigate!
Consider the recurrence relation
\begin{equation*}
a_n = 5a_{n-1} - 6a_{n-2}\text{.}
\end{equation*}
What sequence do you get if the initial conditions are \(a_0 = 1\text{,}\) \(a_1 = 2\text{?}\) Give a closed formula for this sequence.
What sequence do you get if the initial conditions are \(a_0 = 1\text{,}\) \(a_1 = 3\text{?}\) Give a closed formula.
What if \(a_0 = 2\) and \(a_1 = 5\text{?}\) Find a closed formula.