Fibonacci Numbers and the Golden Ratio
THE FIBONACCI SEQUENCE Problems for Lecture 1 1 The Fibonacci numbers can be extended to zero and negative indices using the relation Fn = Fn+2 Fn+1 Determine F0 and find a general formula for F nin terms of F Prove your result using mathematical induction 2 The Lucas numbers are closely related to the Fibonacci numbers and satisfy the
Chapter 2 Fibonacci Numbers - MathWorks
Let’s look carefully at fibonacci m It’s a good example of how to create a Matlab function The first line is function f = fibonacci(n) The first word on the first line says fibonacci m is a function, not a script The remainder of the first line says this particular function produces one output result, f, and takes one input argument, n
Institut Denis Poisson
Created Date: 11/3/2005 12:06:00 PM
Research Project: Fibonacci Numbers
Music - Bartok's Dance Suite Fibonacci's Rabbit Problem 7 Choose one of the following extension activities to include in your project: Find flowers images on the net that have a Fibonacci number of petals See how many of the Fibonacci numbers you can find, and display your images on a page with the number of petals
Nombres de Fermat, Mersenne et Fibonacci - WordPresscom
3Nombres de Fibonacci On définit la suite (f n) des nombres de Fibonacci par : 8 >> >> < >> >>: f0 = 0 f1 = 1 f n+2 = f n+1 +f n pour tout n2N Théorème — Pour tout m>n, PGCD(f m;f n) = fPGCD(m;n) Démonstration Le principe est similaire à celui mis en oeuvre pour les nombres de Mersenne 2
Advanced Fibonacci Techniques & Studies
Fibonacci Clusters, High Percentage Reversals (HPR), and Energy Points 25 Fibonacci Precision Using Harmonic Wave ConvergenceTM 27 Get Free EUR/USD PitchFork Set-Ups 28 The “Rolling” Fibonacci, Market Turnover, and Shadow Bands 28 Candle Wick or Candle Close? 32 What is the Deal with the 50 Value? 33 Special Elliott Wave Projection
Anne Larue, « La suite Fibonacci dans la littérature
Anne Larue, « La suite Fibonacci dans la littérature populaire », colloque du Groupe Phi, dir Emmanuel Bouju, ENS Ulm, juin 2009 La suite Fibonacci dans la littérature pour la jeunesse et la littérature populaire Au sein d’un vaste corpus « littérature et mathématiques », je choisis ici
PROBLEME : QUELQUES RESULTATS SUR LA SUITE DE FIBONACCI n
PROBLEME : QUELQUES RESULTATS SUR LA SUITE DE FIBONACCI On définit la suite de Fibonacci (F n)n∈ par : F 0 = 0, F 1 = 1 et ∀n∈ , F n+2 = F n+1 + F n 1) Déterminer la liste des 10 premiers nombres de Fibonacci (de F 1 à F 10) Ecrire un programme Maple permettant de calculer le nième terme de la suite de Fibonacci
SUITES - bagbouton
Le travail préliminaire nous permet d’affirmer que la suite yn est une suite géométrique de raison a r r 1 2 et que la suite zn est une suite géométrique de raison a r r 2 1 On a donc , 11 1 1 1 0 2 n n u ru u ru r¥ n n et , 21 2 1 2 0 1 n
[PDF] trouver les racines d'un polynome de degré 2
[PDF] polynome degré n
[PDF] définition de la mobilisation
[PDF] factoriser un polynome de degré n
[PDF] polynome degré 2
[PDF] phyllotaxie spiralée
[PDF] définition société civile organisée
[PDF] comment expliquer l'abstention électorale
[PDF] mobilisation des civils première guerre mondiale
[PDF] implication des civils premiere guerre mondiale
[PDF] les civils victimes de la premiere guerre mondiale
[PDF] les conditions de vie des civils pendant la seconde guerre mondiale
[PDF] le fibroscope pour voir ? l'intérieur du corps correction
[PDF] exercice corrigé fibre optique ? saut d'indice
Fibonacci Numbers and the
Golden Ratio
Lecture Notes forJeffrey R. Chasnov
The Hong Kong University of Science and TechnologyDepartment of Mathematics
Clear Water Bay, Kowloon
Hong KongCopyright
c○2016-2022 by Jeffrey Robert Chasnov This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/hk/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.Preface
View the promotional video on YouTubeThese are my lecture notes for my online Coursera course,Fibonacci Numbers and the
Golden Ratio
. These lecture notes are divided into chapters called Lectures, and each Lecture corresponds to a video on Coursera. I have also uploaded the Coursera videos to YouTube, and links are placed at the top of each Lecture. Most of the Lectures also contain problems for students to solve. Less experienced students may find some of these problems difficult. Do not despair! The Lectures can be read and watched, and the material understood and enjoyed without actually solving any problems. But mathematicians do like to solve problems and I have selected those that I found to be interesting. Try some of them, but if you get stuck, full solutions can be read in the Appendix. My aim in writing these lecture notes was to place the mathematics at the level of an advanced high school student. Proof by mathematical induction and matrices, however, may be unfamiliar to a typical high school student and I have provided a short and hopefully readable discussion of these topics in the Appendix. Although all the material presented here can be considered elementary, I suspect that some, if not most, of the material may be unfamiliar to even professional mathematicians since Fibonacci numbers and the golden ratio are topics not usually covered in a University course. So I welcome both young and old, novice and experienced mathematicians to peruse these lecture notes, watch my lecture videos, solve some problems, and enjoy the wonders of the Fibonacci sequence and the golden ratio. For your interest, here are the links to my other online courses. If you are studying matrices and elementary linear algebra, have a look atMatrix Algebra for Engineers
If your interests are differential equations, you may want to browseDifferential Equations for Engineers
For a course on multivariable calculus, try
Vector Calculus for Engineers
And for a course on numerical methods, enroll in
Numerical Methods for Engineers
Contents
I Fibonacci: It"s as Easy as1,1,2,311 The Fibonacci sequence22 The Fibonacci sequence redux
4Practice quiz: The Fibonacci numbers
63 The golden ratio
74 Fibonacci numbers and the golden ratio
95 Binet"s formula
11Practice quiz: The golden ratio
14 II Identities, Sums and Rectangles156 The Fibonacci Q-matrix16
7 Cassini"s identity
198 The Fibonacci bamboozlement
21Practice quiz: The Fibonacci bamboozlement
249 Sum of Fibonacci numbers
2510 Sum of Fibonacci numbers squared
27Practice quiz: Fibonacci sums
2911 The golden rectangle
3012 Spiraling squares
32 III The Most Irrational Number3513 The golden spiral36
14 An inner golden rectangle
3915 The Fibonacci spiral
42Practice quiz: Spirals
44iv