Golden-ratio - Mathematics - Relationship To Fibonacci Sequence

Relationship To Fibonacci Sequence

The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ....

The closed-form expression (known as Binet's formula, even though it was already known by Abraham de Moivre) for the Fibonacci sequence involves the golden ratio:

F\left(n\right) = {{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}} = {{\varphi^n-(-\varphi)^{-n}} \over {\sqrt 5}}.

The golden ratio is the limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence), as originally shown by Kepler:

Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:

\sum_{n=1}^{\infty}|F(n)\varphi-F(n+1)| = \varphi.

More generally:

where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when .

Furthermore, the successive powers of φ obey the Fibonacci recurrence:

\varphi^{n+1} = \varphi^n + \varphi^{n-1}.

This identity allows any polynomial in φ to be reduced to a linear expression. For example:

\begin{align} 3\varphi^3 - 5\varphi^2 + 4 & = 3(\varphi^2 + \varphi) - 5\varphi^2 + 4 \\ & = 3 - 5(\varphi + 1) + 4 \\ & = \varphi + 2 \approx 3.618. \end{align}

However, this is no special property of φ, because polynomials in any solution x to a quadratic equation can be reduced in an analogous manner, by applying:

for given coefficients a, b such that x satisfies the equation. Even more generally, any rational function (with rational coefficients) of the root of an irreducible nth-degree polynomial over the rationals can be reduced to a polynomial of degree n ‒ 1. Phrased in terms of field theory, if α is a root of an irreducible nth-degree polynomial, then has degree n over, with basis .

Read more about this topic:  Golden-ratio, Mathematics

Famous quotes containing the words relationship to, relationship and/or sequence:

    “... the Wall became a magnet for citizens of every generation, class, race, and relationship to the war perhaps because it is the only great public monument that allows the anesthetized holes in the heart to fill with a truly national grief.”
    —Adrienne Rich (b. 1929)

    “Strange and predatory and truly dangerous, car thieves and muggers—they seem to jeopardize all our cherished concepts, even our self-esteem, our property rights, our powers of love, our laws and pleasures. The only relationship we seem to have with them is scorn or bewilderment, but they belong somewhere on the dark prairies of a country that is in the throes of self-discovery.”
    —John Cheever (1912–1982)

    “It isn’t that you subordinate your ideas to the force of the facts in autobiography but that you construct a sequence of stories to bind up the facts with a persuasive hypothesis that unravels your history’s meaning.”
    —Philip Roth (b. 1933)