Exercise 8

This exercise is a continuation of the previous one.

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

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

2. Explain why $$L_n = \left(\frac{1+\sqrt{5}}{2}\right)^n + \left(\frac{1-\sqrt{5}}{2}\right)^n\text{.}$$

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 $L_{20}$.