Written Assignment 2 Fibonacci Sequence

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1 Written Assignment Fibonacci Sequence Banana Slug (Put Your Name here) University of California at Santa Cruz Santa Cruz, CA 9064 USA September, 018 (Write your own abstract.) Abstract 1 Introduction. Who is Fibonacci? (Give a historical account of a person after whom Fibonacci sequence was named. When and where he lived, what he did, etc.) Fibonacci Sequence (You provide proofs for the following statements. You can add or delete comments between any two statements. You can also reorganize the following statements. But supply proofs to all the statements below.) The Fibonacci sequence {a n } n N is given by a 1 = a = 1, a n = a n 1 + a n, for n 3. The first twenty terms of a n are given as follows. 1, 1,, 3,, 8, 13, 1, 34,, 89, 144, 33, 377, 610, 987, 197, 84, 4181, 676. We notice that even integers appear every third term, and every fifth term is a multiple of. These are no accidents. In fact, using the recursive relation above several times, we can quickly deduce the following congruence relations. Lemma.1. The following congruence relations hold for Fibonacci numbers. (See class notes.) a n+3 a n mod, and a n+ 3a n mod. 1

2 Upon inspection of the first five terms, we obtain the following corollary. Corollary.. In the Fibonacci sequence, a n is even if and only if 3 n. Similarly, a n if and only if n. These results can be generalized in the following Theorem. Theorem.3. For m 1 and n 1, a mn is divisible by a m. To prove this result, we need to prepare a Lemma. Lemma.4. For m and n 1, Fibonacci sequence satisfies a m+n = a m 1 a n + a m a n+1. We prove the above identity for a fixed m by strong induction on n 1. First, we verify the formula for n = 1,. When n = 1, the identity takes the form (R.H.S.) = a m 1 a 1 + a m a = a m 1 + a m = a m+1 = (L.H.S.). This verifies the formula for n = 1. For n =, using a = 1, a 3 =, we get a m 1 a + a m a 3 = a m 1 + a m = (a m 1 + a m ) + a m = a m+1 + a m = a m+. Next, assume the formula for n = 1,,..., k with k and prove the formula for n = k + 1. (Complete the proof. You can use the formula for n = k 1, k to deduce the formula for n = k + 1.) Theorem can now be proved by induction on n using the following formula which is a consequence of the above Lemma. a m(k+1) = a mk+m = a mk 1 a m + a mk a m+1. Proof of Theorem. For a fixed m 1, we do induction on n to prove the divisibility property in Theorem. Next, we discuss some summation properties of Fibonacci sequences. results can be proved by simple induction. The following Proposition.. The Fibonacci sequence {a m } has the following properties. a 1 + a 3 + a + + a n 1 = a n, a + a 4 + a a n = a n+1 1, a 1 a + a 3 a ( 1) n+1 a n = 1 + ( 1) n+1 a n 1.

3 In 1843, a French mathematician Binet found the formula for the general term a n of the Fibonacci sequence. Let α and β be two solutions to x x 1 = 0. Thus, α = 1 +, β = 1. Note that α and β satisfy an equation similar to the recursive relation of the Fibonacci sequence a n = a n 1 + a n. Namely, α = α + 1 and β = β + 1. Binet s formula is given in the next Theorem. Theorem.6 (Binet). The general term a n of the Fibonacci sequence is given by [( a n = αn β n α β = ) n ( 1 ) n ]. (We discussed a proof in class. However, there are many different proofs. You can explore them on your own.) The above formula can be used to directly prove various identities involving Fibonacci numbers for which induction may not be easily applied. For example, we can show the following. Proposition.7. The Fibonacci sequence satisfy the following identity. a n+ a n = a n+. Some of you may have noticed that α = 1+ is the golden ratio. This number naturally appears in the Fibonacci sequence as the limit of the successive quotient. The following formula can be proved using Binet s formula by noticing that β n converges to 0 as n goes to infinity. Proposition.8. For the Fibonacci sequence {a n }, we have a n+1 lim = α = 1 +. n a n 3 Fibonacci Sequence in Nature (Do some research and find how Fibonacci sequence appear in nature in various patterns exhibited by plants, creatures, etc. For example, lilies have 3 petals, buttercups have petals, marigolds have 13, etc. Spiral patterns of seeds on the sunflower head has to do with Fibonacci numbers, so is the pattern on pineapples. Find out more on patterns in nature and its relationship to Fibonacci numbers.) (Delete below in your final paper, except possibly the reference. To cite a reference, use the following control sequence [1] to cite the first reference. See the tex file to see how this was actually done.) 3

4 4 Writing Guidelines From this section on, you can delete it in your finished paper. These sections are here as your guidance. 1. Follow all the writing guidelines from your textbook (See Chapter 1) and those discussed in class.. Your paper will be graded for the correctness of the mathematical arguments that you present, the clarity and quality of your exposition and the overall structure of the paper. Give careful thought to the order in which the parts of your paper appear. The above only gives bare minimum materials. Flesh out your paper. Give whatever materials you wish to add, and you will earn more points. 3. Try hard to eliminate typos. Proof read your paper several times. You may lose substantial points due to typos. To that end, do not rely solely on your spell checker; a sentence can be spelled correctly yet still be incorrect grammatically or have a different meaning than you intend. For instance, Give careful thought to the order in which the parts of you paper appear. has a typo that a spell checker won t catch. Collaboration: YES Plagiarism: NO You are allowed (in fact encouraged) to collaborate on this assignment. BUT you need to write your own paper. And YOU need to write it. See integrity /undergraduate students/resources.html for information about what constitutes cheating/plagiarism. Here is a well-written primer on plagiarism and how to avoid it written by UCSC faculty Gregory S. Gilbert (Environmental Studies) and Ingrid M. Parker (Ecology and Evolutionary Biology): 6 Closing Remarks Good luck with learning L A TEX and with your writing. Remember that both Jacqui and I are available in office hours and section for questions. See the class web site for office hour details as well as for additional resources for using L A TEX and writing Mathematics papers. I also want to take the opportunity to tell you that learning L A TEX will take some time but is absolutely worth it, particularly if you expect to continue to use mathematical notation in your future work. L A TEX is simply the best way to represent mathematical formulae, and once you have learned the basics, it is really quite easy to use. You will get better at it quickly, plus it s free as opposed to other typesetting programs. 4

5 References [1] J. Abhau, C.-F. Bödigheimer and R. Ehrenfried, Homology of the mapping class group Γ,1 for surfaces of genus with a boundary curve, arxiv: [] M. F. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math. No 68 (1988), [3] C.-F. Bödigheimer and U. Tillmann, Stripping and splitting decorated mapping class groups, Birkhäuser, Progress in Math. 196(001), [4] K.S. Brown, Cohomology of Groups, Graduate Texs in Mathematics, 87, Springer Verlag, New York (198). [] D. Chataur and L. Menichi, String topology of classifying spaces, arxiv: [6] M. Chas and D. Sullivan, String topology, CUNY, to appear in Ann. of Math. (1999). math.gt/ [7] R. Cohen and V. Godin, A polarized view of string topology, Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser. Vol 308, Cambridge Univ. Press, Cambridge, 004