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Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same ...
Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers: [42] + = + =. In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates φ {\displaystyle \varphi } .
The ratio between two consecutive elements converges to the golden ratio, except in the case of the sequence which is constantly zero and the sequences where the ratio of the two first terms is (). The sequence can be written in the form
A Fibonacci spiral approximates the golden spiral using quarter-circle arcs inscribed in squares derived from the Fibonacci sequence. A golden spiral with initial radius 1 is the locus of points of polar coordinates (,) satisfying = /, where is the golden ratio.
The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers: = = = + + + + + + + +. Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.
In the Fibonacci sequence, each number is the sum of the previous two numbers. Fibonacci omitted the "0" and first "1" included today and began the sequence with 1, 2, 3, ... . He carried the calculation up to the thirteenth place, the value 233, though another manuscript carries it to the next place, the value 377.
For large values of n, the (1 + √ 2) n term dominates this expression, so the Pell numbers are approximately proportional to powers of the silver ratio 1 + √ 2, analogous to the growth rate of Fibonacci numbers as powers of the golden ratio. A third definition is possible, from the matrix formula
Fibonacci search has an average- and worst-case complexity of O(log n) (see Big O notation). The Fibonacci sequence has the property that a number is the sum of its two predecessors. Therefore the sequence can be computed by repeated addition. The ratio of two consecutive numbers approaches the Golden ratio, 1.618... Binary search works by ...