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To see the relation between the sequence and these constants,{{Sfn | Ball | 2003 | pp = 155–156}} note that <math>\varphi</math> and <math>\psi</math> are both solutions of the equation <math display=inline>x^2 = x + 1</math> and thus <math>x^n = x^{n-1} + x^{n-2},</math> so the powers of <math>\varphi</math> and <math>\psi</math> satisfy the Fibonacci recursion. In other words,
<math display=block>\begin{align}
\varphi^n &= \varphi^{n-1} + \varphi^{n-2}, \\[3mu]
\psi^n &= \psi^{n-1} + \psi^{n-2}.
\end{align}</math>
It follows that for any values {{mvar|a}} and {{mvar|b}}, the sequence defined by
<math display=block>U_n=a \varphi^n + b \psi^n</math>
satisfies the same recurrence,
<math display=block>\begin{align}
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<math display=block>
a = \frac{1}{\varphi-\psi} = \frac{1}{\sqrt 5},\quad b = -a,
</math>
producing the required formula.
Taking the starting values {{math|''U''<sub>0</sub>}} and {{math|''U''<sub>1</sub>}} to be arbitrary constants, a more general solution is:
<math display=block> U_n = a\varphi^n + b\psi^n </math>
where
<math display=block>\begin{align}
a&=\frac{U_1-U_0\psi}{\sqrt 5}, \\[3mu]
b&=\frac{U_0\varphi-U_1}{\sqrt 5}.
\end{align}</math>
=== Computation by rounding ===
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=== Limit of consecutive quotients ===
[[Johannes Kepler]] observed that the ratio of consecutive Fibonacci numbers [[convergent sequence|converges]]. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio
<math display=block>\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.</math>
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* [[Mario Merz]] included the Fibonacci sequence in some of his artworks beginning in 1970.{{sfn|Livio|2003|p=176}}
* [[Joseph Schillinger]] (1895–1943) developed [[Schillinger System|a system of composition]] which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature.{{sfn|Livio|2003|p=193}} See also {{slink|Golden ratio|Music}}.
* In [[software development]], Fibonacci numbers are often used by [[Agile management|agile]] teams operating under the [[Scrum (software development)|Scrum]] framework to size their [[product backlog]] items.<ref>{{cite web |last1=Kathuria |first1=Madhur |title=A Guide to Using the Fibonacci Sequence in Scrum |url=https://resources.scrumalliance.org/Article/guide-using-fibonacci-sequence-scrum |website=Scrum Alliance |access-date=8 August 2025}}</ref>
== See also ==
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