The Fibonacci numbers form the sequence of numbers that begins \(0, 1, 1, 2, 3, 5, 8, 13, \ldots\text{.}\) If we let \(F_n\) denote the \(n^{th}\) Fibonacci number, then

\begin{equation*} F_0 =0, F_1 =1, F_2 =1, F_3 =2, F_4 =3,\ldots \text{.} \end{equation*}

In general, a Fibonacci number is the sum of the previous two Fibonacci numbers; that is, \(F_{n+2} = F_{n}+F_{n+1}\) so that we have

\begin{equation*} \begin{aligned} F_{n+2} \amp {}={} F_{n} + F_{n+1} \\ F_{n+1} \amp {}={} F_{n+1}\text{.} \\ \end{aligned} \end{equation*}

  1. If we write \(\xvec_n = \twovec{F_{n+1}}{F_n}\text{,}\) find the matrix \(A\) such that \(\xvec_{n+1} = A\xvec_n\text{.}\)

  2. Show that \(A\) has eigenvalues

    \begin{equation*} \begin{aligned} \lambda_1 \amp {}={} \frac{1+\sqrt{5}}{2}\approx 1.61803\ldots \\ \lambda_2 \amp {}={} \frac{1-\sqrt{5}}{2}\approx -0.61803\ldots \\ \end{aligned} \end{equation*}

    with associated eigenvectors \(\vvec_1=\twovec{\lambda_1}{1}\) and \(\vvec_2=\twovec{\lambda_2}{1}\text{.}\)

  3. Classify this dynamical system as one of the six types that we have seen in this section. What happens to \(\xvec_n\) as \(n\) becomes very large?

  4. Write the initial vector \(\xvec_0 = \twovec{1}{0}\) as a linear combination of eigenvectors \(\vvec_1\) and \(\vvec_2\text{.}\)

  5. Write the vector \(\xvec_n\) as a linear combinations of \(\vvec_1\) and \(\vvec_2\text{.}\)

  6. Explain why the \(n^{th}\) Fibonacci number

    \begin{equation*} F_{n} = \frac{1}{\sqrt{5}}\left[ \left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right]\text{.} \end{equation*}
  7. Use this relationship to compute \(F_{20}\text{.}\)

  8. Explain why \(F_{n+1}/F_{n}\approx \lambda_1\) when \(n\) is very large.

The number \(\lambda_1=\frac{1+\sqrt{5}}{2} = \phi\) is called the golden ratio and is one of mathematics' special numbers.