Let us recall that the famous Fibonacci sequence: $0,1,1,2,3,5,8,13,21,\dots$ is defined as follows:

we put $\phi_0 = 0, \phi_1 = 1$ and define $\phi_{n+2} = \phi_{n+1} + \phi_n$. We want to find a formula for $\phi_n$. To do this

a) Take a 2×2 matrix $A=\begin{pmatrix}1&1\\1&0\end{pmatrix}$ such that

$\begin{pmatrix}\phi_{n+2}\\ \phi_{n+1}\end{pmatrix} = \begin{pmatrix}1&1\\1&0\end{pmatrix} \begin{pmatrix}\phi_{n+1}\\ \phi_n\end{pmatrix}$

b) Diagonalize $A$ and find a formula for $A^n$.

c) Find a formula for $\phi_n$. (You will need to compute an inverse and perform multiplication here).

`d) Show that the vector $(\phi_{n+1} /\phi_n , 1)^T$ converges to an eigenvector of $A$.

What do you think, is it a coincidence?

***for part b) I diagonalized in a way that was not very clean. How do you do c) and d)?

  • $\begingroup$ Sorry I dont know how to make matrix notation but if someone could tell me how I can fix how this problem looks! Since it clearly doesnt make sense now $\endgroup$ May 28, 2016 at 6:41
  • $\begingroup$ You seem to be using a lot of unicode characters, potentially copy-pasting from a PDF. Please consider using the formatting guidelines for math-content posts and format your posts with LaTeX. math.stackexchange.com/editing-help#latex (or use @probablyme 's link, which is more verbose and helpful) $\endgroup$
    – Axoren
    May 28, 2016 at 6:41
  • 1
    $\begingroup$ Formatting tips here. If you copied from somewhere, it didn't come out right. $\endgroup$
    – Em.
    May 28, 2016 at 6:41
  • $\begingroup$ Matrices here. $\endgroup$
    – Em.
    May 28, 2016 at 6:42
  • $\begingroup$ Do you want it to read $\begin{pmatrix}\phi_{n+2}\\ \phi_{n+1}\end{pmatrix}=\begin{pmatrix}1&1\\1&0\end{pmatrix}\begin{pmatrix}\phi_{n+1}\\ \phi_n\end{pmatrix}$? $\endgroup$
    – almagest
    May 28, 2016 at 6:50


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