The Fibonacci Sequence

Transcription

1 Elvis Numbers Elvis the Elf skips up a flight of numbered stairs, starting at step 1 and going up one or two steps with each leap Along with an illustrious name, Elvis parents have endowed him with an obsessivecompulsive disorder Elvis wants to count how many ways he can reach the n th step, which he calls the n th Elvis number, E n Can you figure out the numerical value of E n? (Hint: it s a well-known sequence) If you re stuck, try to work out what E 1, E 2, E 3, E 4 and E 5 are, and guess a pattern When you've found the answer, or have ended up truly stumped in the effort, flip to The Solution to Elvis the Elf's Eccentric Exercise (p 61) 58

2 Courtesy of Columbia University Library Historical Digression Fibonacci: The Greatest European Mathematician of the Middle Ages Leonardo of Pisa is widely acknowledged to have been the greatest European mathematician of the Middle Ages He is known to us today as Fibonacci, or son of Bonaccio The son of an Italian merchant, Leonardo studied under a Muslim teacher and made travels to Egypt, Syria, and Greece As a result, he picked up much of the mathematics that was being disseminated through centers of learning in the medieval Muslim world; in particular, he became a key proponent of Hindu- Arabic numerals, from which were derived the digits 1 through 9 Leonardo da Pisa (Fibonacci) c Leonardo s classic book, Liber Abaci (Book of the Abacus), completed in 1202, discusses numerous arithmetic algorithms, often explained through problems Certainly the most famous problem in Liber Abaci is: How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on? By drawing a tree diagram (see the next page) showing the number of pairs of rabbits at the end of each month, we discover that the number of pairs of rabbits in the first six months are given by the following sequence: 1, 1, 2, 3, 5, 8, This famous sequence, named the Fibonacci sequence in honor of Leonardo of Pisa, has the property that each term is the sum of the two previous terms 1 In the tree diagram on the following page, we mark the first pair of rabbits with the number 1 Since they do not produce offspring in the first two months, there is only one pair at the end of the second month On the first day of the third month, they give birth to another pair (labelled 2) so there are two pairs at the end of the third month The arrows in the tree diagram show the offspring in the successive months up to the end of the sixth month 1 There is actually a journal, The Fibonacci Quarterly, devoted specifically to the Fibonacci numbers 59

3 Historical Digression In the diagram The below, Fibonacci denotes Rabbit the first Problem pair of rabbits, denotes an adult pair of rabbits, and denotes a newborn pair of rabbits End of Month The Fibonacci Rabbit Problem Number of Pairs Sequences are often defined by an equation relating the n th term to the previous terms In such a case, we say that the sequence is defined recursively The Fibonacci sequence can be defined recursively by the equation, F n = F n 1 + F n 2 where F n denotes the n th term of the sequence Can you see why the number of pairs of rabbits in the n th month should satisfy this relation? 60

4 The Solution to Elvis the Elf s Eccentric Exercise Elvis the Elf s eccentric exercise was posed in Elvis Numbers (p 58) The Fibonacci numbers F 1, F 2, F 3, are defined as follows F 1 = F 2 = 1 F n = F n 1 + F n 2 for n 3 Q E D The n th Elvis number is actually the n th Fibonacci number! How does Elvis see this? Elvis begins by laboriously counting the number of ways of climbing to the n th step in his green suede shoes: or or or E 1 = 1 E 2 = 1 E 3 = 2 E 4 = 3 To avoid intense boredom, Elvis thinks of a way to work out E 5 without doing lots of counting (and exercise) Rather than skipping up the stairs as is his wont, Elvis steps carefully up one at a time On the n th step, he writes down E n He thinks, How can I get to the fifth step? Well, I can get to the third step, and then jump up two steps There are E 3 ways of doing that Then again, I might get to the fourth step, and then jump up one step There are E 4 ways of doing that But I have to do exactly one of those two things, so the number of ways I can get to step 5 is E 3 + E 4 Thus E 5 = E 4 + E 3 The final jump to step 5 is from step 3 or step 4 Elvis quickly generalizes this procedure to show that E n = E n 1 + E n 2 for n > 2 Along with E 1 = 1 and E 2 = 1, this is enough to determine E n in general But the Fibonacci numbers are given by the same recipe (F 1 = 1 and F 2 = 1, and F n = F n 1 + F n 2 ), so these two sequences must be the same! QED? E 4 E 3 E 2 E 1 61

5 Food for Thought How many ways can you spell ELVIS? Start at the E and move either down or to the right at each stage E L V I S L V I S V I S I S S The answer and a further mystery is given on page 65 Hint: What is this puzzle doing at the end of this section? 62

6 A Formula for the n th Term of the Fibonacci Sequence A few pages earlier, we observed that the Fibonacci numbers, F n, are defined according to the following rules: F 1 = F 2 = 1 F n = F n 1 + F n 2 for n 3 Then, remarkably, it can be shown that F n is given by the following equation, often called Binet s Formula: Teachers! n n Give your students Binet s formula Invite them to guess the sequence it s F = n quite a surprise to find nice Fibonacci 5 numbers coming out of that hideous formula! Then they can try to prove it This is quite a mouthful! We can express F n more simply using some constants that we ll pull out of our hat: τ = , σ = τ and σ are Greek letters, called tau and sigma respectively Tau (which rhymes with Ow! ) is a constant known to the ancient Greeks as the golden mean It has a habit of turning up in the oddest places in mathematics; some of them are described in the chapter Fibonacci & The Golden Mean In terms of τ and σ: n n τ σ F = n 5 There are several other ways of expressing F n that you might prefer Since σ = 1 n τ (Check this!), F n τ = n 1 τ 5 σ can be expressed in terms of τ in a second way: σ = 1 τ (Check this too!) Thus ( ) n n τ 1 τ F = n 5 63

7 We know from the definition of the Fibonacci sequence (although not from the formula for its general term) that F n is a positive integer for all n It follows from n n τ σ σ n 1 the equation above that F = n is an integer for all n Since < for all n (verify this for yourself), τ n 5 is closer to F n than it is to any other integer That is, F n is the integer closest to τ n 5 Since σ n 5 becomes very small as n gets large, τ n 5 gets very very close to F n (Try it on your calculator with n = 201 ) You can prove that the crazy formula for F n actually works by employing the same method we used in the Solution to Elvis the Elf s Eccentric Exercise (p 61) the method of mathematical induction (although induction wasn t actually mentioned by name in that section) Basically, induction is a great way of proving things when you already know (or have guessed) what the answer is Using the magical formula, or induction, or a lot of ingenuity, it is possible to prove some remarkable results about Fibonacci numbers, which appear in Funny Fibonacci Facts (p 150) 1 It is indeed remarkable that for any large n, τ n / 5 is very very close to an integer Even more remarkable is that for any large n, τ n is also very very close to an integer try it on a calculator and see! These oddities, and others, come up in Some Interesting Properties of the Golden Mean (p 155) 64

8 Answer to Food for Thought on Page 62 There are 16 ways to spell Elvis Is it a coincidence that the answer turned out to be a power of 2? As a hint, find out how many ways there are to spell NO in the following diagram, starting with the N in the upper left-hand corner, and moving either down or to the right N O O If you re not convinced (and you re probably not), then count the number of ways of spelling NEGATIVE : N E G A T I V E E G A T I V E G A T I V E A T I V E T I V E I V E V E E If you are still not convinced, read the upside-down hint below One strategy we can apply to this problem is to ask, How many paths from N to an E on the diagonal if we may only move to the right or down? From the N, we can reach an adjacent E in 2 ways From each of those E's we can reach a G in 2 ways Thus there are 4 different paths from the N to a G From each G, we can reach an A in two ways, so there are 8 paths from the N to an A Continuing this pattern, we find there are 2 7 = 128 paths from N to an E on the diagonal Here is another question: there are eight E's in the diagonal above How many paths are there ending at each particular E? Answer: How and why does this relate to Pascal's Triangle (discussed in The Triangles of Pascal, Chu Shih-Chieh, and Sierpinski, p 100)? 65

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