Difference between revisions of "Fibonacci sequence"
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The Fibonacci sequence can be written [[recursion|recursively]] as <math>F_n=F_{n-1}+F_{n-2}</math>. | The Fibonacci sequence can be written [[recursion|recursively]] as <math>F_n=F_{n-1}+F_{n-2}</math>. | ||
| − | + | == Phi == | |
| − | == | ||
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]]. | 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]]. | ||
| − | + | == Binet's formula == | |
| − | == | ||
'''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}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)</math>. | It is <math>\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)</math>. | ||
| − | {{ | + | ==Problems== |
| + | === Introductory === | ||
| + | === Intermediate === | ||
| + | # Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle. <div style="text-align:right">(Manhattan Mathematical Olympiad 2004)</div> | ||
| + | # A [[fair]] coin is to be tossed <math>10_{}^{}</math> times. Let <math>i/j^{}_{}</math>, in lowest terms, be the [[probability]] that heads never occur on consecutive tosses. Find <math>i+j_{}^{}</math>. <div style="text-align:right">([[1990 AIME Problems/Problem 9|1990 AIME, Problem 9]])</div> | ||
| + | === Olympiad === | ||
==See also== | ==See also== | ||
* [[Combinatorics]] | * [[Combinatorics]] | ||
| + | |||
| + | {{stub}} | ||
Revision as of 17:05, 4 March 2007
The Fibonacci sequence is a sequence of integers in which the first and second term are both equal to 1, and each subsequent term is the sum of the two preceding it. The first few terms are 
.
The Fibonacci sequence can be written recursively as 
.
Contents
Phi
Ratios between successive terms, 
, 
, 
, 
, 
, tend towards the limit phi.
Binet's formula
Binet's formula is an explicit formula used to find any nth term.
It is 
.
Problems
Introductory
Intermediate
- Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle. (Manhattan Mathematical Olympiad 2004)
- A fair coin is to be tossed
times. Let 
, in lowest terms, be the probability that heads never occur on consecutive tosses. Find 
.
times. Let 
, in lowest terms, be the 
. Olympiad
See also
This article is a stub. Help us out by expanding it.
