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Fractional Evaluation of Golden Ratio

Tyler Berezowsky

 

July 29, 2014

The golden ratio (Φ) is equivalent to the number approach when the Fibonacci sequence is
performed indefinitely and the last two terms are divided. The Fibonacci sequence can be
described as follows:

Fn+2 =  Fn+1 + Fn (1)

The sequence traditionally begins with zero and one, but can began with any numbers and Φ
will be approached. Starting the sequence with zero and one an example of the series is
below:

01123581321345589144

Various methods can be used to evaluate Φ the following is the fractional method.

             1 Φ = 1 + -------1--         1 + 1+ 1+1...-

Note that Φ = 1 + 1Φ,

 2 Φ  - Φ - 1 = 0

    ∘  ----------       √-- 1-±----1 --4-(--1) = 1 ±--5-=  1.618...         2              2
Obviously, none of the bitmaps linked. Crap.