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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 } .
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 ...
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.
All Fibonacci-like integer sequences appear in shifted form as a row of the Wythoff array; the Fibonacci sequence itself is the first row and the Lucas sequence is the second row. Also like all Fibonacci-like integer sequences, the ratio between two consecutive Lucas numbers converges to the golden ratio.
Inspired by a similar Stolarsky array previously defined by Stolarsky (1977), Morrison (1980) defined the Wythoff array as follows. Let = + denote the golden ratio; then the th winning position in Wythoff's game is given by the pair of positive integers (⌊ ⌋, ⌊ ⌋), where the numbers on the left and right sides of the pair define two complementary Beatty sequences that together include ...
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.
A golden triangle. The ratio a/b is the golden ratio φ. The vertex angle is =.Base angles are 72° each. Golden gnomon, having side lengths 1, 1, and .. A golden triangle, also called a sublime triangle, [1] is an isosceles triangle in which the duplicated side is in the golden ratio to the base side: