Consider the sequence of numbers given by
\begin{eqnarray}
a_0 & = & 1, \\
a_1 & = & 1, \\
a_n & = & 3a_{n-1} - 2a_{n-2}~\text{ for } n \geq 2.
\end{eqnarray}
What is the value of \(a_{1107}\)?
The answer is \(1\).
In fact, \(a_n = 1\) for all \(n \geq 2\).
To see this, notice that \(a_n = 1\)
if \(a_{n-1} = a_{n-2} = 1\) since \(3(1) - 2(1) = 3 - 2 = 1\).
But \(a_0 = a_1 = 1\). So \(a_2 = 1\). This in turn gives
\(a_3 = 1\), and so on.