Nov 22, 2016 in this expository paper written to commemorate fibonacci day 2016, we discuss famous relations involving the fibonacci sequence, the golden ratio, continued fractions and nested radicals, and show how these fit into a more general framework stemming from the quadratic formula. Binets formula expresses the n th fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive fibonacci numbers tends to the golden ratio as n increases. Fundamentals of a new kind of mathematics based on the golden. Fibonacci and lucas numbers the fibonacci quarterly. Fibonacci numbers are defined by the recurrence relation there exist a lot of properties about fibonacci numbers. View homework help fibonacci numbers and the golden ratio from math 101 at excelsior college. This shows the relationship between fibonacci numbers and golden ratio. The absence of even perfect numbers in the associated pell sequence follows from the paper 12 by prasad and rao. He exploited results on exponential diophantine equations and some of the ideas implicit in f. The fibonacci and lucas numbers and have only the singular point. The golden ratio is a number found by dividing a line into two parts so that the longer part is divided. Since the second half of 20th century, golden section and fibonacci numbers.
Fibonacci and the golden mean university of georgia. Some golden ratio generalized fibonacci and lucas sequences. Dec 19, 2014 dual fibonacci and dual lucas numbers are defined with dual fibonacci and lucas quaternions in nurkan and g. Three identities concerning fibonacci and lucas numbers re. This page intentionally left blank fibonacci and lucas numbers with. Fibonacci numbers, the golden section and the golden string section appears in nature. In the following section we investigate some products of fibonacci numbers and lucas numbers. Some geometry related to the golden section 14 full text. Fibonacci and lucas numbers with applications pdf free download. A study on dual hyperbolic fibonacci and lucas numbers. Fibonacci and lucas numbers, and the golden section. In particular, there is a beautiful combinatorial identity to fibonacci numbers from, filipponi introduced the incomplete fibonacci numbers and the incomplete lucas numbers.
Various characteristics of the ordinary fibonacci and lucas sequences, many known for centuries, are shown to be common to generalized sequences related to the golden ratio. Lucas demonstration 7, 8 that there are no perfect fibonacci or lucas numbers. What is the golden ratio and how is it related to the. The lucas numbers are formed in the same way as the fibonacci numbers by adding the latest two to get the next, but instead of starting at 0 and 1 fibonacci numbers the lucas number series starts with 2 and 1.
Identities involving lucas or fibonacci and lucas numbers as. Pdf identification whether a certain number belongs to either fibonacci or lucas. The fibonacci numbers and golden section in nature 1. In this paper, we define a new generalization of the fibonacci and lucas pnumbers. A new approach to fibonacci, lucas numbers and dual vectors. A list of the first 100 fibonacci and lucas numbers 83 full text. Also like all fibonaccilike integer sequences, the ratio between two consecutive lucas numbers converges to the golden ratio.
Unit digits rightenddigits, modular rings, golden ratio, fibonacci and lucas numbers, pythagorean triples. Fibonacci numbers and the golden section this is a lesson plan about the fibonacci sequence and the golden ratio, suitable for both middle school first part and high school math students. May 15, 2012 the golden ratio and fibonacci numbers by r. This text examines meta fibonacci numbers, proceeding to a survey of the golden section in the plane and space.
The international club of the golden section, 6 mccreary trail, bolton, on, canada l7e 2c8. Ask your students, how do they think this sequence of numbers continues. Fibonacci numbers and the golden section homeschool math. The fibonacci principle of forming a new number by an appropriate combination of previous numbers has been extended to yield sequences with surprising and sometimes mystifying properties. Based on general form of fibonacci sequence, elements of the lucas sequence can be generated using the equation 2. Though the results can be established by induction method as well, binets formula is mainly used to prove all of them. The lucas numbers are closely related to the fibonacci numbers and satisfy the same recursion. We also define new vectors which are called dual fibonacci vectors. The links below go to a fantastic website about fibonacci numbers and the golden ratio there is lots and lots more to learn. Three identities concerning fibonacci and lucas numbers. In 4, luca established the absence of perfect numbers in the fibonacci and lucas sequences. The fibonacci series, the lucas series and golden ratio. Factoring fibonacci numbers factoring fibonacci and lucas numbers has been carried out on a large scale. A mathematical history of the golden number unabridged by roger herzfischles.
On the intersections of fibonacci, pell, and lucas numbers. A similar interpretation exists for lucas sequence. Hundreds of fibonacci and lucas identities, as well as identities involving. Azarian as the ratio of two consecutive lucas numbers both converge to the famous golden ratio. See more ideas about golden ratio, fibonacci spiral and divine proportion. On the products of fibonacci numbers and lucas numbers. Then try the puzzles, mathematical explorations, and learn about applications of fibonacci numbers. Zthat includes sequences of fibonacci, pell, lucas, and lucaspell numbers. It has been called the golden mean, the golden section, the golden cut, the divine proportion, the fibonacci number and the mean of fhidias and has a value of 1. This, the first, looks at the fibonacci numbers and why they appear in various family trees and patterns of spirals of leaves and seeds. In this paper, we define a new generalization of the fibonacci and lucas p numbers. Studying about the fibonacci sequence and the golden ratio makes an excellent project for high school to write a report on.
The interested reader is referred to hardy and wright 1 for more about. The second column shows the ratios of successive terms. Pdf relationship between fibonacci and lucas sequences and. Fibonacci numbers are strongly related to the golden ratio. In this study, we work on these dual fibonacci and dual lucas numbers. Fibonacci numbers and the golden ratio 3 in the next section we will use this formula to show how is related to the fibonacci sequence. This is a thorough and easytofollow study of the fibonacci numbers and their close relatives the lucas numbers and the generalized fibonacci numbers all satisfy the same recursion but have different initial values. Hyperbolic fibonacci and lucas functions, golden fibonacci. Relationship between fibonacci and lucas sequences and their. It includes fun activities that really help students see patterns in nature.
The irrational number, golden ratio 5 which is also known as golden section by the ancient greeks 1, golden proportion, mean ratio, divine proportion, golden mean or golden number 6, is widely used in modern sciences particularly in theoretical physics 7. This golden ratio had been used by egyptians in the construction of their great pyramids. Fibonacci numbers and the golden ratio department of. Lucas numbers is found by computing the sum of the previous two terms. Fibonacci and lucas numbers are determined identically with the symmetric hyperbolic fibonacci and lucas functions as follows. Now with a geometers sketchpad dynamic demonstration.
Docagnes, cassini, catalan, negadual fibonacci identities, binet formulas and relations of them. The fibonacci sequence, lucas numbers and their generalization have many interesting properties and applications to almost every field. The site also provides excellent explanations and activities about the golden section and golden string that also appear in nature. Lets define some properties of the fibonacci and lucas numbers. It has been called the golden mean, the golden section, the golden cut, the divine proportion, the fibonacci number and the mean of. Fibonacci numbers and the golden ratio the golden ratio is. The golden section and the fibonacci quadratic equation 9. Twobytwo matrices related to the fibonacci numbers 62 full text. In the literature, there are many identities about fibonacci and lucas numbers. Fibonacci numbers are named after italian mathematician leonardo of pisa, later known as.
The fibonacci sequence is a source of many nice and interesting identities. Fibonacci numbers, the golden section and the golden string. The other two sequences coxeter mentions above have other pairs of starting values but then proceed with the exactly the same rule. That is, conjecture for any positive integer n, the fibonacci numbers satisfy. Fibonacci numbers converges to give approximates of 1. It has been of interest to mathematicians, physicists, philosophers, architects, artists and even m since antiq. Web pages of current records are maintained by blair kelly. Moreover, the intersection in these cases also forms a lucas sequence. Combining equations 1 and 2, we arrive at the sum of fibonacci numbers with alternating. Fibonacci numbers and the golden ratio the golden ratio. May 05, 2010 but the numbers in fibonaccis sequence have a life far beyond rabbits, and show up in the most unexpected places. Starting with any two distinct positive numbers, and forming a sequence using the fibonacci rule, the ratios of consecutive terms will always approach the golden ratio. There are a modest number of typographical errors, none of them confusing.
Vajda, fibonacci and lucas numbers, and the golden section. Research article on the products of fibonacci numbers and. Volume 02 issue 04, july 20 fibonacci numbers and golden. The golden section and the fibonacci quadratic equation 9 full text.
Further, we build up the tree diagrams for generalized fibonacci and lucas psequence and derive the recurrence. Fibonacci numbers fn have been completely factored for. If we take the ratio of two successive numbers in fibonaccis series, 1, 1, 2, 3, 5, 8, and. The second page then examines why the golden section is used by nature in some detail, including animations of growing plants. The puzzling world of fibonacci numbers a pair of pages with plenty of playful problems to perplex the professional and the parttime puzzler. Research article on the products of fibonacci numbers and lucas numbers bijendrasingh,kiransisodiya,andfarooqahmad school of studies in mathematics, vikram university ujjain, india. Then youre ready to study where all the golden section is found. Sums of powers of fibonacci and lucas numbers minnesota. Both, fibonacci and lucas sequences can be used for. Fibonacci, lucas, generalised fibonacci and golden section. Fibonacci resources properties of fibonacci numbers and generalised sequences. In this study, we give three identities concerning fibonacci and lucas. The fibonacci and lucas numbers and do not have branch points and branch cuts over the complex plane. Facts and conjectures about factorizations of fibonacci and.
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