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The mathematics of the golden ratio and of the Fibonacci sequence are intimately interconnected. The Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, .... The closed-form expression for the Fibonacci sequence involves the golden ratio: F ( n ) = n ( 1 ) n 5 = n ( ) n 5 . {\displaystyle 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: lim n F ( n + 1 ) F ( n ) = . {\displaystyle \lim _{n\to \infty }{\frac {F(n+1)}{F(n)}}=\varphi .} Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates ; e.g., 987/610 1.6180327868852. These approximations are alternately lower and higher than , and converge on as the Fibonacci numbers increase, and: n = 1 | F ( n ) F ( n + 1 ) | = . {\displaystyle \sum _{n=1}^{\infty }|F(n)\varphi -F(n+1)|=\varphi .} More generally: lim n F ( n + a ) F ( n ) = a , {\displaystyle \lim _{n\to \infty }{\frac {F(n+a)}{F(n)}}={\varphi }^{a},} where above, the ratios of consecutive terms of the Fibonacci sequence, is a case when a = 1 {\displaystyle a=1} . Furthermore, the successive powers of obey the Fibonacci recurrence: n + 1 = n + n 1 . {\displaystyle \varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}.} This identity allows any polynomial in to be reduced to a linear expression. For example: 3 3 5 2 + 4 = 3 ( 2 + ) 5 2 + 4 = 3 [ ( + 1 ) + ] 5 ( + 1 ) + 4 = + 2 3.618. {\displaystyle {\begin{aligned}3\varphi ^{3}-5\varphi ^{2}+4&=3(\varphi ^{2}+\varphi )-5\varphi ^{2}+4\\&=3[(\varphi +1)+\varphi ]-5(\varphi +1)+4\\&=\varphi +2\approx 3.618.\end{aligned}}} The reduction to a linear expression can be accomplished in one step by using the relationship k = F k + F k 1 , {\displaystyle \varphi ^{k}=F_{k}\varphi +F_{k-1},} where F k {\displaystyle F_{k}} is the kth Fibonacci number. However, this is no special property of , because polynomials in any solution x to a quadratic equation can be reduced in an analogous manner, by applying: x 2 = a x + b {\displaystyle x^{2}=ax+b} 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 is a root of an irreducible nth-degree polynomial, then Q ( ) {\displaystyle \mathbb {Q} (\alpha )} has degree n over Q {\displaystyle \mathbb {Q} } , with basis { 1 , , , n 1 } {\displaystyle \{1,\alpha ,\dots ,\alpha ^{n-1}\}} .

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