Difference between revisions 3684376 and 3684378 on mswiki

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[[Image:FibonacciBlocks.svg|thumb|180px|right|Suatu ubinan dengan segi empat yang tepinya adalah nombor Fibonaci berturut-turut pada panjangnya]]
(contracted; show full)

establishing the base cases of the induction, proving that
:<math>F(n)={{\varphi^n-(1-\varphi)^n} \over {\sqrt 5}}</math> for all <math> n\, .</math>

Therefore, for any two starting values, a combination <math>a,b</math> can be found such that the function <math>F_{a,b}(n)\,</math> is the exact closed formula for the series.

====
Computation by rounding====
SincePengiraan melalui pembundaran====
Memandangkan <math>\begin{matrix}|1-\varphi|^n/\sqrt 5 < 1/2\end{matrix}</math> for allbagi semua <math>n\geq 0</math>, the nunombeor <math>F(n)</math> is the closest integer to <math>\varphi^n/\sqrt 5\, .</math> Therefore it can be found by [[Rounding#Rounding_in_an_exact_computation|rounding]], or in terms of the [[floor functionadalah integer yang paling hampir dengan <math>\varphi^n/\sqrt 5\, .</math> Oleh itu, ia boleh didapati dengan [[Pembundaran#Pembundaran dalam pengiraan tepat|pembundaran]], atau dari segi [[fungsi lantai]]:
:<math>F(n)=\bigg\lfloor\frac{\varphi^n}{\sqrt 5} + \frac{1}{2}\bigg\rfloor.</math>

===Limit of consecutive quotients===

[[Johannes Kepler]] observed that the ratio of consecutive Fibonacci numbers 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 the limit approaches the golden ratio <math>\varphi</math>.<ref>{{cite book | last=Kepler | first=Johannes | title=A New Year Gift: On Hexagonal(contracted; show full)
*[http://web.archive.org/web/20070715032716/http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=630&bodyId=1002 Fibonacci Numbers] at [http://web.archive.org/web/20060212072618/http://mathdl.maa.org/convergence/1/ Convergence]
* [http://www.tools4noobs.com/online_tools/fibonacci/ Online Fibonacci calculator]

[[Kategori:Fibonacci numbers|*]]
[[Kategori:Articles containing proofs]]

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