Fibonacci numbers @ HMMT
by t0rajir0u, Feb 21, 2009, 10:11 PM
Today I gave a talk at the Harvard-MIT Math Tournament as a mini-event on the Fibonacci numbers. The slides are available here (~1 MB).
The focus of the talk was the material from this post and this post and, to a lesser extent, this post. Briefly, there are three ways to think about the proof of Binet's formula:
1. The space of sequences satisfying
is ia vector space of dimension 
, and there exists a basis consisting of the two geometric sequences 
.
2. The generating function
has a partial fraction decomposition.
3. The powers of the matrix![$ \left[ \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right]$](//latex.artofproblemsolving.com/2/d/6/2d66c0270096f9afdd4f3ec02fd060e23c13617f.png)
describe the Fibonacci numbers and this matrix is diagonalizable.
As I discussed in some of the posts above, these proofs are essentially the same (although it is the third that leads into the material from this post, which I unfortunately didn't have enough time to cover); each is about relating geometric series to the eigenvectors of a shift operator. See the posts above or the slides for details.
The focus of the talk was the material from this post and this post and, to a lesser extent, this post. Briefly, there are three ways to think about the proof of Binet's formula:
1. The space of sequences satisfying
is ia vector space of dimension 
, and there exists a basis consisting of the two geometric sequences 
.2. The generating function
has a partial fraction decomposition.3. The powers of the matrix
![$ \left[ \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right]$](http://latex.artofproblemsolving.com/2/d/6/2d66c0270096f9afdd4f3ec02fd060e23c13617f.png)
describe the Fibonacci numbers and this matrix is diagonalizable.As I discussed in some of the posts above, these proofs are essentially the same (although it is the third that leads into the material from this post, which I unfortunately didn't have enough time to cover); each is about relating geometric series to the eigenvectors of a shift operator. See the posts above or the slides for details.
April 2009
March 2009