Editing Golden ratio (section)

===Relationship to Fibonacci and Lucas numbers===
{{further|Fibonacci number#Relation to the golden ratio}}
{{see also|Lucas number#Relationship to Fibonacci numbers}}

{{multiple image|align=right|direction=vertical|image1=Fibonacci Spiral.svg|image2=Lucas number spiral.svg|width=250|footer=A [[Fibonacci sequence|Fibonacci spiral]] (top) which approximates the [[golden spiral]], using [[Fibonacci number|Fibonacci sequence]] square sizes up to {{math|21}}. A different approximation to the golden spiral is generated (bottom) from stacking squares whose lengths of sides are numbers belonging to the sequence of [[Lucas numbers]], here up to {{math|76}}.}}

[[Fibonacci number]]s and [[Lucas number]]s have an intricate relationship with the golden ratio. In the Fibonacci sequence, each term <math>F_n</math> is equal to the sum of the preceding two terms <math>F_{n-1}</math> and <math>F_{n-2}</math>, starting with the base sequence {{tmath|0,1}} as the 0th and 1st terms <math>F_0</math> and <math>F_1</math>:

{{bi |left=1.6 |1={{tmath|0,}} {{tmath|1,}} {{tmath|1,}} {{tmath|2,}} {{tmath|3,}} {{tmath|5,}} {{tmath|8,}} {{tmath|13,}} {{tmath|21,}} {{tmath|34,}} {{tmath|55,}} {{tmath|89,}} {{tmath|\ldots}} ({{OEIS2C|id=A000045}}).}}

The sequence of Lucas numbers (not to be confused with the generalized [[Lucas sequence]]s, of which this is part) is like the Fibonacci sequence, in that each term <math>L_n</math> is the sum of the previous two terms <math>L_{n-1}</math> and <math>L_{n-2}</math>, however instead starts with {{tmath|2,1}} as the 0th and 1st terms <math>L_0</math> and <math>L_1</math>:

{{bi |left=1.6 |1={{tmath|2,}} {{tmath|1,}} {{tmath|3,}} {{tmath|4,}} {{tmath|7,}} {{tmath|11,}} {{tmath|18,}} {{tmath|29,}} {{tmath|47,}} {{tmath|76,}} {{tmath|123,}} {{tmath|199,}} {{tmath|\ldots}} ({{OEIS2C|id=A000032}}).}}

Exceptionally, the golden ratio is equal to the [[limit of a sequence|limit]] of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers:<ref name=tattersall />
<math display=block>
\lim_{n\to\infty} \frac{F_{n+1}}{F_n}
= \lim_{n\to\infty} \frac{L_{n+1}}{L_n}
= \varphi.
</math>

In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates {{tmath|\varphi}}. For example,

{{bi |left=1.6 |1=<math>\frac{F_{16}}{F_{15}} = \frac{987}{610} = 1.6180327\ldots\ </math> and <math>\ \frac{L_{16}}{L_{15}} = \frac{2207}{1364} = 1.6180351\ldots.</math> }}

These approximations are alternately lower and higher than {{tmath|\varphi}}, and converge to {{tmath|\varphi}} as the Fibonacci and Lucas numbers increase.

[[Closed-form expression]]s for the Fibonacci and Lucas sequences that involve the golden ratio are:

<math display=block>
F\left(n\right)
 = \frac{\varphi^n - (-\varphi)^{-n}}{\sqrt5}
 = \frac{\varphi^n - (1 - \varphi)^n}{\sqrt5}
 = \frac{1}{\sqrt5}\left[\left({ 1+ \sqrt{5} \over 2}\right)^n - \left({ 1- \sqrt{5} \over 2}\right)^n\right],
 </math>
<math display=block>
L\left(n\right)
 = \varphi^n + (- \varphi)^{-n}
 = \varphi^n + (1 - \varphi)^n
 = \left({ 1+ \sqrt{5} \over 2}\right)^n + \left({ 1- \sqrt{5} \over 2}\right)^n .
 </math>

Combining both formulas above, one obtains a formula for {{tmath|\textstyle \varphi^n}} that involves both Fibonacci and Lucas numbers:
<math display=block>
\varphi^n = \tfrac12\bigl(L_n + F_n \sqrt{5}~\!\bigr).
</math>

Between Fibonacci and Lucas numbers one can deduce {{tmath|1=\textstyle L_{2n} = 5 F_n^2 + 2(-1)^n = L_n^2 - 2(-1)^n}}, which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the [[square root of five]]:
<math display=block>\lim_{n\to\infty} \frac{L_n}{F_n}=\sqrt{5}.</math>

Indeed, much stronger statements are true:
<math display=block>\begin{align}
& \bigl\vert L_n - \sqrt5 F_n \bigr\vert = \frac{2}{\varphi^n} \to 0, \\[5mu]
& \bigl(\tfrac12 L_{3n}\bigr)^2 = 5 \bigl(\tfrac12 F_{3n}\bigr)^2 + (-1)^n.
\end{align}</math>

These values describe {{tmath|\varphi}} as a [[fundamental unit (number theory)|fundamental unit]] of the [[algebraic number field]] {{tmath|\mathbb{Q}\bigl(\sqrt5~\!\bigr)}}.

Successive powers of the golden ratio obey the Fibonacci [[recurrence relation|recurrence]], {{tmath|1=\textstyle \varphi^{n+1} = \varphi^n + \varphi^{n-1} }}.

The reduction to a linear expression can be accomplished in one step by using:
<math display=block>
\varphi^n = F_n \varphi + F_{n-1}.
</math>

This identity allows any polynomial in {{tmath|\varphi}} to be reduced to a linear expression, as in:

<math display=block>\begin{align}
3\varphi^3 - 5\varphi^2 + 4
&= 3(\varphi^2 + \varphi) - 5\varphi^2 + 4 \\[5mu]
&= 3\bigl((\varphi + 1) + \varphi\bigr) - 5(\varphi + 1) + 4 \\[5mu]
&= \varphi + 2 \approx 3.618033.
\end{align}</math>

Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by [[Series (mathematics)|infinite summation]]:
<math display=block>\sum_{n=1}^{\infty}\bigl|F_n\varphi-F_{n+1}\bigr| = \varphi.</math>

In particular, the powers of {{tmath|\varphi}} themselves round to Lucas numbers (in order, except for the first two powers, {{tmath|\textstyle \varphi^0}} and {{tmath|\varphi}}, are in reverse order):

<math display=block>\begin{align}
\varphi^0 &= 1, \\[5mu]
\varphi^1 &= 1.618033989\ldots \approx 2, \\[5mu]
\varphi^2 &= 2.618033989\ldots \approx 3, \\[5mu]
\varphi^3 &= 4.236067978\ldots \approx 4, \\[5mu]
\varphi^4 &= 6.854101967\ldots \approx 7,
\end{align}</math>

and so forth.<ref name=parker4d /> The Lucas numbers also directly generate powers of the golden ratio; for {{tmath|n \ge 2}}:
<math display=block>
\varphi^n = L_n - (- \varphi)^{-n}.
</math>

Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of ''third'' consecutive Fibonacci numbers equals a Lucas number, that is {{tmath|1=\textstyle L_n = F_{n-1} + F_{n+1}\! }}; and, importantly, that {{tmath|1=\textstyle L_n F_n = F_{2n}\! }}.

Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the [[golden spiral]] (which is a special form of a [[logarithmic spiral]]) using quarter-circles with radii from these sequences, differing only slightly from the ''true'' golden logarithmic spiral. ''Fibonacci spiral'' is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.