On the intersections of fibonacci, pell, and lucas numbers. Then try the puzzles, mathematical explorations, and learn about applications of fibonacci numbers. Further, we build up the tree diagrams for generalized fibonacci and lucas psequence and derive the recurrence. 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. The golden section and the fibonacci quadratic equation 9 full text. Then youre ready to study where all the golden section is found. This shows the relationship between fibonacci numbers and golden ratio. 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. It includes fun activities that really help students see patterns in nature. A mathematical history of the golden number unabridged by roger herzfischles. It has been called the golden mean, the golden section, the golden cut, the divine proportion, the fibonacci number and the mean of.
In the literature, there are many identities about fibonacci and lucas numbers. Now with a geometers sketchpad dynamic demonstration. 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. What is the golden ratio and how is it related to the. The second column shows the ratios of successive terms. Since the second half of 20th century, golden section and fibonacci numbers.
Relationship between fibonacci and lucas sequences and their. Fundamentals of a new kind of mathematics based on the golden. A new approach to fibonacci, lucas numbers and dual vectors. In this study, we give three identities concerning fibonacci and lucas. In particular, there is a beautiful combinatorial identity to fibonacci numbers from, filipponi introduced the incomplete fibonacci numbers and the incomplete lucas numbers. Sums of powers of fibonacci and lucas numbers minnesota. Zthat includes sequences of fibonacci, pell, lucas, and lucaspell numbers. Fibonacci and the golden mean university of georgia. This, the first, looks at the fibonacci numbers and why they appear in various family trees and patterns of spirals of leaves and seeds. On the products of fibonacci numbers and lucas numbers. Though the results can be established by induction method as well, binets formula is mainly used to prove all of them. Based on general form of fibonacci sequence, elements of the lucas sequence can be generated using the equation 2. 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.
The fibonacci sequence is a source of many nice and interesting identities. The fibonacci numbers and golden section in nature 1. Fibonacci numbers are defined by the recurrence relation there exist a lot of properties about fibonacci numbers. May 15, 2012 the golden ratio and fibonacci numbers by r. A similar interpretation exists for lucas sequence. Both, fibonacci and lucas sequences can be used for. Hyperbolic fibonacci and lucas functions, golden fibonacci. 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. The site also provides excellent explanations and activities about the golden section and golden string that also appear in nature. In this paper, we define a new generalization of the fibonacci and lucas p numbers.
The absence of even perfect numbers in the associated pell sequence follows from the paper 12 by prasad and rao. Fibonacci numbers and the golden section homeschool math. In this paper, we define a new generalization of the fibonacci and lucas pnumbers. There are a modest number of typographical errors, none of them confusing. Some golden ratio generalized fibonacci and lucas sequences. Ask your students, how do they think this sequence of numbers continues. Fibonacci numbers, the golden section and the golden string. Identities involving lucas or fibonacci and lucas numbers as. Lets define some properties of the fibonacci and lucas numbers. A list of the first 100 fibonacci and lucas numbers 83 full text.
Some geometry related to the golden section 14 full text. 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. Three identities concerning fibonacci and lucas numbers re. The puzzling world of fibonacci numbers a pair of pages with plenty of playful problems to perplex the professional and the parttime puzzler. 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. Pdf identification whether a certain number belongs to either fibonacci or lucas. Twobytwo matrices related to the fibonacci numbers 62 full text. Unit digits rightenddigits, modular rings, golden ratio, fibonacci and lucas numbers, pythagorean triples.
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. Hundreds of fibonacci and lucas identities, as well as identities involving. 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. Fibonacci numbers converges to give approximates of 1. This golden ratio had been used by egyptians in the construction of their great pyramids.
Docagnes, cassini, catalan, negadual fibonacci identities, binet formulas and relations of them. The interested reader is referred to hardy and wright 1 for more about. This text examines meta fibonacci numbers, proceeding to a survey of the golden section in the plane and space. Pdf relationship between fibonacci and lucas sequences and. Lucas demonstration 7, 8 that there are no perfect fibonacci or lucas numbers. Studying about the fibonacci sequence and the golden ratio makes an excellent project for high school to write a report on.
The links below go to a fantastic website about fibonacci numbers and the golden ratio there is lots and lots more to learn. Dec 19, 2014 dual fibonacci and dual lucas numbers are defined with dual fibonacci and lucas quaternions in nurkan and g. See more ideas about golden ratio, fibonacci spiral and divine proportion. Fibonacci numbers and the golden ratio department of. May 05, 2010 but the numbers in fibonaccis sequence have a life far beyond rabbits, and show up in the most unexpected places. Fibonacci numbers fn have been completely factored for. Combining equations 1 and 2, we arrive at the sum of fibonacci numbers with alternating. Fibonacci and lucas numbers, and the golden section. Also like all fibonaccilike integer sequences, the ratio between two consecutive lucas numbers converges to the golden ratio. Fibonacci numbers, the golden section and the golden string section appears in nature. The golden ratio is a number found by dividing a line into two parts so that the longer part is divided. Fibonacci and lucas numbers the fibonacci quarterly. The other two sequences coxeter mentions above have other pairs of starting values but then proceed with the exactly the same rule.
The fibonacci sequence, lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci numbers and the golden ratio the golden ratio. Fibonacci numbers are strongly related to the golden ratio. The fibonacci and lucas numbers and have only the singular point. Facts and conjectures about factorizations of fibonacci and. The fibonacci series, the lucas series and golden ratio. The golden section and the fibonacci quadratic equation 9. The international club of the golden section, 6 mccreary trail, bolton, on, canada l7e 2c8. That is, conjecture for any positive integer n, the fibonacci numbers satisfy.
We also define new vectors which are called dual fibonacci vectors. Fibonacci, lucas, generalised fibonacci and golden section. Lucas numbers is found by computing the sum of the previous two terms. A study on dual hyperbolic fibonacci and lucas numbers. Fibonacci and lucas numbers are determined identically with the symmetric hyperbolic fibonacci and lucas functions as follows. Selected identities involving the fibonacci and lucas numbers 52 full text. Vajda, fibonacci and lucas numbers, and the golden section. 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. The lucas numbers are closely related to the fibonacci numbers and satisfy the same recursion.
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. Fibonacci numbers are named after italian mathematician leonardo of pisa, later known as. Fibonacci resources properties of fibonacci numbers and generalised sequences. The second page then examines why the golden section is used by nature in some detail, including animations of growing plants. In this study, we work on these dual fibonacci and dual lucas numbers. Moreover, the intersection in these cases also forms a lucas sequence. The fibonacci and lucas numbers and do not have branch points and branch cuts over the complex plane. View homework help fibonacci numbers and the golden ratio from math 101 at excelsior college. Volume 02 issue 04, july 20 fibonacci numbers and golden. Factoring fibonacci numbers factoring fibonacci and lucas numbers has been carried out on a large scale. In 4, luca established the absence of perfect numbers in the fibonacci and lucas sequences. It has been of interest to mathematicians, physicists, philosophers, architects, artists and even m since antiq.
Research article on the products of fibonacci numbers and. If we take the ratio of two successive numbers in fibonaccis series, 1, 1, 2, 3, 5, 8, and. Research article on the products of fibonacci numbers and lucas numbers bijendrasingh,kiransisodiya,andfarooqahmad school of studies in mathematics, vikram university ujjain, india. In the following section we investigate some products of fibonacci numbers and lucas numbers. Azarian as the ratio of two consecutive lucas numbers both converge to the famous golden ratio.
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