Definition 5.4. Fibonacci Sequence is an integral sequence defined by Fo=0, F₁ = 1, Fn = Fn-1 + Fn-2 26 The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, ... The closed form expression for the nth Fibonacci number is Fn = on-(-)-n √5 where is the golden ratio. Thus we see that continued fractions connects the golden ratio to the Fibonacci sequence. The convergents to the continued fraction of the golden ratio are precisely the ratio of consecutive Fibonacci numbers.
Definition 5.4. Fibonacci Sequence is an integral sequence defined by Fo=0, F₁ = 1, Fn = Fn-1 + Fn-2 26 The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, ... The closed form expression for the nth Fibonacci number is Fn = on-(-)-n √5 where is the golden ratio. Thus we see that continued fractions connects the golden ratio to the Fibonacci sequence. The convergents to the continued fraction of the golden ratio are precisely the ratio of consecutive Fibonacci numbers.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 1E
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Show how continued fractions connect the golden ratio to the Fibonacci sequence
Transcribed Image Text:Definition 5.4. Fibonacci Sequence is an integral sequence defined by
Fo=0, F₁ = 1, Fn = Fn-1 + Fn-2
26
The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, ...
The closed form expression for the nth Fibonacci number is
Fn
=
on-(-)-n
√5
where is the golden ratio. Thus we see that continued fractions connects the golden ratio
to the Fibonacci sequence. The convergents to the continued fraction of the golden ratio are
precisely the ratio of consecutive Fibonacci numbers.
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