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MathDoctorBob and Fibonacci

I ran into an interesting page which purported to be about various ways of looking at (or deriving) the trigonometric identity

`sin`^{2} + cos^{2} = 1

but the site has lots of stuff including the trail I'm following here, which involves the Fibonacci numbers. Strang derives the Binet formula for the n^{th} Fibonacci number, F_{n}, using linear algebra:

`F`_{n} = 1/√5 [φ_{1}^{n} - φ_{2}^{n}]

where `φ`_{1} is the golden mean or golden ratio, and

`φ`_{1} + φ_{2} = 1

The approach is very nice, because it uses the eigenvalues and eigenvectors of the matrix

`[ 1 1 ]`

[ 1 0 ]

which involves solving `λ`^{2} - λ -1 = 0, whose solutions are `φ`_{1} and φ_{2}. That is, `φ`_{1} and `φ`_{2} are eigenvalues of that matrix.

However, I got lost in the middle of Strang's version (in the book), and cast about for another explanation. I found a great one here.

This video turns out to be one of a large number by MathDoctorBob. I am very impressed with the quality of these videos, based on the first half-dozen or so that I've looked at.

Finally, back at the first site, we use two limits (as `n => ∞`):

`φ`_{1}^{n} => ∞

φ_{2}^{n} = (1 - φ_{1})^{n} => 0

and the Binet formula to get a limit for the ratio with large n:

`F`_{n+1} / F_{n} = φ