Recall from class that we found that the fibonacci sequence, with $f_0 = 0$, $f_1 = 1$ and $f_n = f_{n - 2} + f_{n - 1}$, had a closed form $$f_n = \frac{1}{\sqrt{5}} \left( \phi^n - \widehat{\phi}^n \right),$$ where $$\phi = \frac{1 + \sqrt{5}}{2} \; \text{and} \; \widehat{\phi} = \frac{1 - \sqrt{5}}{2}.$$

the lucas numbers are defined in the same way, but with different starting values. let $l_0$ be the zeroth lucas number and $l_1$ be the first. if
\begin{align*}
l_0 & = 2 \\
l_1 & = 1 \\
l_n & = l_{n - 1} + l_{n - 2} \; \text{for}\; n \geq 2
\end{align*}
then what is the tenth lucas number? (note: we seek a numerical answer.)