Difference between revisions of "Fibonacci sequence"

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The '''Fibonacci sequence''' is a series of numbers in which each number is the sum of the two preceding it (the first two terms are simply 1).  The first few terms are <math>1,1,2,3,5,8,13,21,34,55,...</math>.  Ratios between successive terms, <math>\frac{1}{1}</math>, <math>\frac{2}{1}</math>, <math>\frac{3}{2}</math>, <math>\frac{5}{3}</math>, <math>\frac{8}{5}</math>, tend towards the limit [[phi]].
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The '''Fibonacci sequence''' is a series of numbers in which each number is the sum of the two preceding it (the first two terms are simply 1).  The first few terms are <br><math>1,1,2,3,5,8,13,21,34,55,...</math>.   
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The Fibonacci sequence can be written recursively as <math>F_n=F_{n-1}+F_{n-2}</math>.
 
The Fibonacci sequence can be written recursively as <math>F_n=F_{n-1}+F_{n-2}</math>.
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== Introduction ==
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Ratios between successive terms, <math>\frac{1}{1}</math>, <math>\frac{2}{1}</math>, <math>\frac{3}{2}</math>, <math>\frac{5}{3}</math>, <math>\frac{8}{5}</math>, tend towards the limit [[phi]].
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== Intermediate ==
  
 
'''Binet's formula''' is an explicit formula used to find any nth term.
 
'''Binet's formula''' is an explicit formula used to find any nth term.
 
It is <math>\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)</math>
 
It is <math>\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)</math>

Revision as of 17:31, 20 June 2006

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding it (the first two terms are simply 1). The first few terms are
$1,1,2,3,5,8,13,21,34,55,...$.

The Fibonacci sequence can be written recursively as $F_n=F_{n-1}+F_{n-2}$.


Introduction

Ratios between successive terms, $\frac{1}{1}$, $\frac{2}{1}$, $\frac{3}{2}$, $\frac{5}{3}$, $\frac{8}{5}$, tend towards the limit phi.


Intermediate

Binet's formula is an explicit formula used to find any nth term. It is $\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)$