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Fibonacci sequence the Golden Ratio

The Fibonacci sequence is the sequence defined by:

F0=0, F1=1, Fn=Fn1+Fn2,  for  n=2,3,4,....F_0 = 0,  F_1 = 1 ,  F_n = F_{n−1 }+ F_{n−2} \textrm{,  for }  n = 2, 3, 4, . . . .

 

First 15 sequence values:

n01234567891011121314
Fn01123581321345589144233377

 

The number

φ=1+52=1.61803... φ = \frac{1 + \sqrt{5}} {2} = 1.61803 . . .

 

is called the Golden Ratio. It satisfies the property

φ=1+1φ φ = 1 + \frac{1}{φ}

 

Can you show that the sequence of ratio of successive Fibonacci numbers

Fn+1Fn \frac{F_{n+1}}{F_n}

 

converges to the Golden Ratio?

limnFn+1Fn=φ \lim_{n\to\infty} \frac{F_{n+1}}{F_n} = φ

 

Observe that

F14/F13=377/2331.618025 F_{14} /F_{13} = 377/233 ≈ 1.618025

 

is already pretty close to φ.

What conditions must be met for this ratio to converge?

Fibonacci Spiral

Extension: Missing Squares

An elegant variation of this idea makes use of two rectangles of such proportion that they fit side by side to make a perfect 8 by 8 checkerboard. When the pieces are rearranged to make the larger rectangle, there is an apparent gain in area of one square unit.  Where did the extra square come from, and what does this have to do with Fibonacci?

fibonacci squares