##### Exercise8

This exercise is a continuation of the previous one.

The Lucas numbers $$L_n$$ are defined by the same relationship as the Fibonacci numbers: $$L_{n+2}=L_{n+1}+L_n\text{.}$$ However, we begin with $$L_0=2$$ and $$L_1=1\text{,}$$ which leads to the sequence $$2,1,3,4,7,11,\ldots\text{.}$$

1. As before, form the vector $$\xvec_n=\twovec{L_{n+1}}{L_n}$$ so that $$\xvec_{n+1}=A\xvec_n\text{.}$$ Express $$\xvec_0$$ as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{,}$$ eigenvectors of $$A\text{.}$$

2. Explain why

\begin{equation*} L_n = \left(\frac{1+\sqrt{5}}{2}\right)^n + \left(\frac{1-\sqrt{5}}{2}\right)^n\text{.} \end{equation*}
3. Explain why $$L_n$$ is the closest integer to $$\phi^n$$ when $$n$$ is large, where $$\phi = \lambda_1$$ is the golden ratio.

4. Use this observation to find $$L_{20}\text{.}$$

in-context