Editing Fibonacci sequence (section)

=== 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>\varphi\colon </math> <ref>{{Citation|last=Kepler |first=Johannes |title=A New Year Gift: On Hexagonal Snow |year=1966 |isbn=978-0-19-858120-8 |publisher=Oxford University Press |page= 92}}</ref><ref>{{Citation | title = Strena seu de Nive Sexangula | year = 1611}}</ref>
<math display=block>\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\varphi.</math>

This convergence holds regardless of the starting values <math>U_0</math> and <math>U_1</math>, unless <math>U_1 = -U_0/\varphi</math>. This can be verified using [[#Binet's formula|Binet's formula]]. For example, the initial values 3 and 2 generate the sequence 3, 2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, ...&thinsp;. The ratio of consecutive elements in this sequence shows the same convergence towards the golden ratio.

In general, <math>\lim_{n\to\infty}\frac{F_{n+m}}{F_n}=\varphi^m
</math>, because the ratios between consecutive Fibonacci numbers approaches <math>\varphi</math>.

: [[File:Fibonacci tiling of the plane and approximation to Golden Ratio.gif|thumb|upright=2.2|left|Successive tilings of the plane and a graph of approximations to the golden ratio calculated by dividing each Fibonacci number by the previous]]
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