I n an arihmeic sequence, a common difference separaes each erm in he sequence. Using funcion noaion, every arihmeic sequence is a linear funcion of he form y = f(x) = ax + b (where he domain is he posiive inegers and a is he common difference). In a geomeric sequence, you calculae each successive erm by muliplying by he same number. Using funcion noaion, every geomeric sequence is an exponenial funcion of he form y = f(x) = ar (where he domain again is he posiive inegers and r is he com- x 1 mon raio or muliplier). In a power sequence, we calculae each successive erm by raising consecuive posiive inegers o he same power. The nex sequence of imporance is he recursive sequence. Arihmeic and geomeric sequences can be undersood using recursion mehods. Recursion is a mehod 1 where, based upon a saring erm, you repeaedly applying he same arihmeic operaion o each erm o arrive a he subsequen erm. In recursion, you mus know he value of he erm immediaely before he erm you are rying o find. A recursive formula always has wo pars. Firs, we choose he saring value or 1 (he firs erm or sub 1 ). Second, a recursion formula for n ( sub n ) as a funcion of n 1 ( sub n 1 ), he erm preceding n. Here is a simple example. Le 1 = 5. Our formula is n = 5 n 1. This is all we need o generae he sequence: 5, 25, 625, 3125, In his example, we have creaed a geomeric sequence (common raio or muliplier is 5). Term 1 2 3 4 Value 5 25 625 3125 Nex, we le 1 = 5 and our formula n = n 1 + 5. Our sequence: 5, 10, 15, 20, (an arihmeic sequence!). Term 1 2 3 4 Value 5 10 15 20 One of he more famous sequences ha we can undersand by recursive mehods is he Fibonacci sequence, a sequence relaed o he growh of populaions. In 1788, European selers brough he rabbi o he coninen of Ausralia. Now, he wild rabbi populaion is over 200,000,000 (compared o he human populaion of Ausralia of 20,000,000). Rabbis have no naural predaors and, because of heir rapid populaion growh, hey do millions of dollars of damage o Ausralian agriculure every year. I was he sudy of he growh of rabbi populaions ha led a 13 h cenury European mahemaician o he discovery of he Fibonacci sequence. 2 He was Leonardo of Pisa (1170-1240), also known as Leonardo Fibonacci (son of Bonaccio, his faher). Here is he sequence: Public Domain 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 1 We also call hese mehods algorihms, a se of rules for solving a problem in a finie number of seps. The word is Arabic coming from algorism, a Lainizaion of he Arab mahemaician al-khwarizmi (ca. 780-ca. 850), in imporan figure in he developmen of algebra. 2 The Fibonacci sequence was known in ancien India, where i was applied o he merical (poery)sciences (prosody), long before i was known in Europe. Developmens have been aribued o Pingala (200 BC), Virahanka (6 h cenury AD), Gopāla (ca. 1135 AD), and Hemachandra (ca. 1150 AD). 1 of 6
Wha is happening? The firs wo erms are 1, and each successive erm is he sum of he preceding pair of erms. We le F n represen he n h erm of his Fibonacci sequence. In his sequence, we mus know he value of he firs wo erms. F 1 = 1 and F 2 = 1. Wha is he formula? Fn+ 1= Fn+ Fn 1. To work he sequence, we mus sar a n = 2. Noe his able: Term F 1 F 2 F 3 F 4 F 5 F 6 F 7 F 8 F 9 Value 1 1 2 3 5 8 13 21 34 We can beer see wha is going on by sudying his able: n Fn 1 F n F n + 1 2 1 1 2 3 1 2 3 4 2 3 5 5 3 5 8 6 5 8 13 Noice how he erms of he Fibonacci sequence unfold in columns 2, 3, and 4 (labeled Fn 1, F, n and F n + 1respecively). Fibonacci, a raveling merchan, visied Arabia. There he discovered he Hindu-Arabic sysem of numeraion (decimal/posiional noaion). In 1202, he wroe Liber abaci (meaning Book of calculaion ), and by i, inroduced his sysem of numbers o Europe. This book is a collecion of business mahemaics, linear and quadraic equaions, including square roos and cube roos, and oher opics. He also said, These are he nine figures of he Indians 9 8 7 6 5 4 3 2 1. Wih hese nine figures, and wih he symbol 0, which in Arabic is called zephirum [means empy JN], any number can be wrien, as will be demonsraed below. 3 In his book, he wroe abou he resuls of his sudy of rabbi populaions. He waned o know he number of pairs of rabbis produced each year if one begins wih a single pair ha maures during he firs monh and hen produces anoher pair of rabbis afer ha (of course, he is assuming ha no rabbis die in his process!). The Fibonacci sequence answers ha quesion. How? Suppose here is one pair of rabbis in he monh of January ha breed a second pair in he monh of February. Thereafer, hese produce anoher pair monhly; i.e., each pair of rabbis produces anoher pair in he second monh following birh and hereafer one pair per monh. How many pairs will here be afer one year? To solve his problem, we need o abulae four columns: 1. The number of pairs of breeding rabbis a he beginning of a given monh. 2. The number of pairs of non-breeding rabbis a he beginning of he monh. 3. The number of pairs of rabbis bred during he monh. 4. The number of pairs of rabbis living a he end of he monh. 3 Cied in Alfred S. Posamenier, Mah Wonders (Alexandria, VA: Associaion for Supervision and Curriculum Developmen, 2003), p. 32. 2 of 6
Monh 1 Breeding 2 Non-breeding 3 Bred 4 Toal January 0 1 0 1 February 1 0 1 2 March 1 1 1 3 April 2 1 2 5 May 3 2 3 8 June 5 3 5 13 July 8 5 8 21 Augus 13 8 13 34 Sepember 21 13 21 55 Ocober 34 21 34 89 November 55 34 55 144 December 89 55 89 233 Again, noe how he erms of he Fibonacci sequence unfold in he columns labeled 1, 2, 3, and 4. Since he Fibonacci sequence connecs o he growh of cerain populaions, we frequenly find i in he physical creaion, especially of he biological naure. Le s see how i reveals he geneics of he honeybee (haplodiploid reproducion). The drone (male bee) haches from an egg ha has no been ferilized (i has only one paren, is moher). The ferilized egg produces only a female, eiher a queen bee or a worker bee (i has wo parens, a faher and a moher). If we look a jus one generaion, a drone has one paren, a moher. If we look a he second generaion of ancesors, he drone has wo grandparens: he moher and faher of his moher. A he hird generaion, here are hree grea grandparens: he moher and faher of his grandmoher, and he moher (no faher) of his grandfaher. If you figure ou he parenage of his grea grandparens, you will find ha here are five grea, grea grandparens. Taking he drone himself as he saring generaion, he number of bees in each generaion follows he Fibonacci sequence: Generaion Number of Bees 1 1 drone 2 1 moher 3 2 grandparens 4 3 grea grandparens 5 5 grea, grea grandparens 6 8 grea, grea, grea grandparens 7 13 grea, grea, grea, grea grandparens female male 3 of 6
Man, made in he image of God, has used he properies of his sequence in his ar, archiecure, and music. For example, he piano keys shown are for one ocave (a series of a group of 8) of he chromaic scale on he piano. The number of black keys in he ocave is 5 (hese keys used o be called he penaonic scale where pena is Greek for 5), which is he fifh erm of he Fibonacci sequence. The number of whie keys in he ocave is 8 (hese keys are called he major scale), which is he sixh erm of he sequence. The oal number of keys is 13 (he sevenh erm). Noe also ha he black keys are srucured in ses of 2 and 3 (also Fibonacci numbers). Fibonacci numbers are also revealed in he spiral arrangemen of peals, pinecones, and pineapples (check his ou in a grocery sore or wih some pinecones). In he pinecone spiral, here are 5 spirals one way and 8 he oher. In pineapple spirals, he pair one way and he oher can be 5 and 8, or 8 and 13. In he sunflower spirals, he combinaion can be 8 and 13, 21 and 34, 34 and 55, 55 and 89, and 89 and 144. Truly, he Fibonacci sequence is ubiquious in God s creaion! We can compare he growh of power sequences (he sequence of squares and he sequence of cubes) and Fibonacci numbers (much more Source: isockphoo explosive ) by observing his able: Number Square Cube Fibonacci Numbers 1 1 1 F 1 1 2 4 8 F 2 1 3 9 27 F 3 2 4 16 64 F 4 3 5 25 125 F 5 5 6 36 216 F 6 8 7 49 343 F 7 13 8 64 512 F 8 21 9 81 729 F 9 34 10 100 1000 F 10 55 11 121 1331 F 11 89 12 144 1728 F 12 144 13 169 2197 F 13 233 14 196 2744 F 14 377 15 225 3375 F 15 610 16 256 4096 F 16 987 17 289 4913 F 17 1597 18 324 5832 F 18 2584 19 361 6859 F 19 4181 20 400 8000 F 20 6765 21 441 9,261 F 21 10,946 22 484 10,648 F 22 17,711 23 529 12,167 F 23 28,657 24 576 13,824 F 24 46,368 25 625 15,625 F 25 75,025 4 of 6
Number Square Cube Fibonacci Numbers 26 676 17,576 F 26 121,393 27 729 19,683 F 27 196,418 28 784 21,952 F 28 317,811 29 841 24,389 F 29 514,229 30 900 27,000 F 30 832,040 5 of 6 Le s conclude our brief sudy of he Fibonacci sequence by revealing i in a number rick. I wan you o creae a able (10 rows by 2 columns). Pick any wo numbers where each number is a wo-digi number. In my example, I will use 15 and 18 as my saring number. Make sure you choose wo numbers differen from hese. Wrie hese wo numbers in rows 1 and 2. Add hem (e.g., 15 + 18 = 33) and wrie your sum in row 3 (you are engaging he Fibonacci recursion algorihm). Nex, add your sum o he second number you chose (e.g, 18 + 33 = 51) and wrie his sum in row 4. Coninue his process unil you reach row 10. Can you deermine a fas way o deermine he sum of all he numbers in he second column? 1. 15 2. 18 3. 33 4. 51 5. 84 6. 135 7. 219 8. 354 9. 573 10. 927 Sum? We invoke algebra o help us find his sum. Le a and b represen he wo numbers you chose. The following able explains wha you are doing in erms of a and b. In review, noe a + a = 2a and b + 2b = 3b 1. a 2. b 3. a + b 4. a + 2b 5. 2a + 3b 6. 3a + 5b 7. 5a + 8b 8. 8a + 13b 9. 13a + 21b 10. 21a + 34b Sum 55a + 88b Now, sum up all he a s and b s (in algebra, we are summing like erms ) in his able. You should ge 55a + 88b (his represens he sum of all he numbers in he second column). Noe he appearance of he Fibonacci numbers as coefficiens of a and b. Compare his sum wih row 7; i.e., 55a + 88b wih 5a + 8b. The sum is 11 imes he value of row 7! Using he disribuive rule, 11(5a + 8b) = 55a + 88b. To find he sum, all you need o do is muliply row 7 by 11!
In my example where I seleced 15 and 18 as my saring numbers), 219 11 = (use he disribuive rule again) = 219(10 + 1) = 2190 + 219 = 2409! Such is he power of algebra! 6 of 6