1.
Consider the recurrence relation
\begin{equation*}
a_n = 5a_{n-1} - 6a_{n-2}
\text{.}
\end{equation*}
Since the \(n\)th term is given as a combination of the two previous terms, we will need two initial terms to determine the sequence. Different initial terms will give different sequences.
(a)
What sequence do you get if the initial conditions are \(a_0 = 1\text{,}\) \(a_1 = 2\text{?}\) Give the first five terms (including 1 and 2).
(b)
Based on the first few terms, what is a closed formula for this sequence?
\(a_n =\)
(c)
What sequence do you get if the initial conditions are \(a_0 = 1\text{,}\) \(a_1 = 3\text{?}\) Give the first five terms.
(d)
Based on the first few terms, what is a closed formula for this sequence?
\(a_n =\)
(e)
What sequence do you get if the initial conditions are \(a_0 = 2\text{,}\) \(a_1 = 5\text{?}\) Give the first five terms.
(f)
Based on the first few terms, what is a closed formula for this sequence?
\(a_n =\)
Hint.
How do the terms in this sequence relate to the terms in the previous two sequences?
