The Fibonacci Numbers
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- Rudolph Walton
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1 The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in nature, art, music, geometry, mathematics, economics, and more...
2 The Fibonacci Numbers The numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Each Fibonacci number is the sum of the previous two Fibonacci numbers! Definition The n th Fibonacci number is written as F n. F 3 = 2, F 5 = 5, F 10 = 55 We have a recursive formula to find each Fibonacci number: F n = F n 1 + F n 2
3 What makes these numbers so special? They occur many times in nature,
4 What makes these numbers so special? They occur many times in nature,
5 What makes these numbers so special? They occur many times in nature, art,
6 What makes these numbers so special? They occur many times in nature, art, architecture,...
7 Let s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? white Calla Lily
8 Let s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? euphorbia
9 Let s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? trillium
10 Let s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? columbine
11 Let s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? bloodroot
12 Let s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? black-eyed susan
13 Let s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? shasta daisy
14 Let s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? field daisies
15 Let s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? back of passion flower
16 Let s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? front of passion flower
17 Let s look at these Fibonacci numbers in nature. There are exceptions... fuschia
18 Looking more closely at the blossom in the center of the flower: What do we see?
19 Looking more closely at the blossom in the center of the flower: The stamen form spirals:
20 Looking more closely at the blossom in the center of the flower: How many are there?
21 Looking more closely at the blossom in the center of the flower: How many are there?
22 Looking more closely at the blossom in the center of the flower: Not just in daisies. In Bellis perennis:
23 Looking more closely at the blossom in the center of the flower: Not just in daisies. In Bellis perennis:
24 These numbers occur in other plants as well: The seed-bearing leaves of a simple pinecone:
25 These numbers occur in other plants as well: Find and count the number of spirals:
26 These numbers occur in other plants as well: Find and count the number of spirals:
27 These numbers occur in other plants as well: Find and count the number of spirals:
28 These numbers occur in other plants as well: Find and count the number of spirals:
29 Another pinecone: Find and count the number of spirals:
30 Another pinecone: Find and count the number of spirals:
31 Another pinecone: Find and count the number of spirals:
32 In fruits and vegetables, like cauliflower: Find and count the number of spirals:
33 In fruits and vegetables, like cauliflower: Find and count the number of spirals:
34 In fruits and vegetables, like cauliflower: Find and count the number of spirals:
35 In fruits and vegetables, like romanesque: Find and count the number of spirals:
36 In fruits and vegetables, like romanesque: Find and count the number of spirals:
37 In fruits and vegetables, like romanesque: Find and count the number of spirals:
38 In fruits and vegetables, like pineapples: Find and count the number of spirals:
39 In fruits and vegetables, like pineapples: Find and count the number of spirals:
40 In fruits and vegetables, like pineapples: Find and count the number of spirals:
41 In fruits and vegetables, like pineapples: Find and count the number of spirals:
42 In fruits and vegetables, like pineapples: Find and count the number of spirals:
43 The Fibonacci Numbers Let s examine some interesting properties of these numbers. The numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Each Fibonacci number is the sum of the previous two Fibonacci numbers! Let n any positive integer. If F n is what we use to describe the n th Fibonacci number, then F n = F n 1 + F n 2
44 The Fibonacci Numbers For example, let s look at the sum of the first several of these numbers. What is F 1 + F 2 + F 3 + F 4 + F n =???
45 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Let s look at the first few: F 1 + F 2 = = 2 F 1 + F 2 + F 3 = = 4 F 1 + F 2 + F 3 + F 4 = = 7 F 1 + F 2 + F 3 + F 4 + F 5 = = 12 F 1 + F 2 + F 3 + F 4 + F 5 + F 6 = = 20 See a pattern?
46 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... It looks like maybe the sum of the first few Fibonacci numbers is one less another Fibonacci number! But which Fibonacci number? F 1 + F 2 + F 3 + F 4 = = 7 = F 6 1 The sum of the first 4 Fibonacci numbers is one less the 6 th Fibonacci number! F 1 + F 2 + F 3 + F 4 + F 5 + F 6 = = 20 = F 8 1 The sum of the first 6 Fibonacci numbers is one less the 8 th Fibonacci number!
47 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... It looks like the sum of the first n Fibonacci numbers is one less the (n + 2) nd Fibonacci number. That is, F 1 + F 2 + F 3 + F F n = F n+2 1 Let s check the formula, for n = 7: F 1 +F 2 +F 3 +F 4 +F 5 +F 6 +F 7 = = 33 = F 9 1 It s got hope to be true! Let s try to prove it!
48 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Conjecture For any positive integer n, the Fibonacci numbers satisfy: F 1 + F F n = F n+2 1 Let s prove this (and then we ll call it a Theorem.)
49 Trying to prove: F 1 + F 2 + F n = F n+2 1 We know F 3 = F 1 + F 2. So, rewriting this a little: F 1 = F 3 F 2 Also: we know F 4 = F 2 + F 3. So: F 2 = F 4 F 3 In general: F n = F n+2 F n+1
50 Trying to prove: F 1 + F 2 + F n = F n+2 1 F 1 = F 3 F 2 F 2 = F 4 F 3 F 3 = F 5 F 4 F 4 = F 6 F 5.. F n 1 = F n+1 F n F n = F n+2 F n+1 Adding up all the terms on the left sides will give us something equal to the sum of the terms on the right sides.
51 Trying to prove: F 1 + F 2 + F n = F n+2 1 F 1 = F 3 F 2 F 2 = F 4 F 3 F 3 = F 5 F 4 F 4 = F 6 F 5.. F n 1 = F n+1 F n + F n = F n+2 F n+1
52 Trying to prove: F 1 + F 2 + F n = F n+2 1 F 1 = /// F 3 F 2 F 2 = F 4 /// F 3 F 3 = F 5 F 4 F 4 = F 6 F 5.. F n 1 = F n+1 F n + F n = F n+2 F n+1
53 Trying to prove: F 1 + F 2 + F n = F n+2 1 F 1 = /// F 3 F 2 F 2 = /// F 4 /// F 3 F 3 = F 5 /// F 4 F 4 = F 6 F 5.. F n 1 = F n+1 F n + F n = F n+2 F n+1
54 Trying to prove: F 1 + F 2 + F n = F n+2 1 F 1 = /// F 3 F 2 F 2 = /// F 4 /// F 3 F 3 = /// F 5 /// F 4 F 4 = F 6 /// F 5.. F n 1 = F n+1 F n + F n = F n+2 F n+1
55 Trying to prove: F 1 + F 2 + F n = F n+2 1 F 1 = /// F 3 F 2 F 2 = /// F 4 /// F 3 F 3 = /// F 5 /// F 4 F 4 = /// F 6 /// F 5. F n 1 = F n+1 /// F n. + F n = F n+2 F n+1
56 Trying to prove: F 1 + F 2 + F n = F n+2 1 F 1 = /// F 3 F 2 F 2 = /// F 4 /// F 3 F 3 = /// F 5 /// F 4 F 4 = /// F 6 /// F 5. F n 1 = F////// n+1 /// F n + F n = F n+2 F////// n+1.
57 Trying to prove: F 1 + F 2 + F n = F n+2 1 F 1 = /// F 3 F 2 F 2 = /// F 4 /// F 3 F 3 = /// F 5 /// F 4 F 4 = /// F 6 /// F 5. F n 1 = F////// n+1 /// F n + F n = F n+2 F////// n+1. F 1 + F 2 + F F n = F n+2 F 2
58 Trying to prove: F 1 + F 2 + F n = F n+2 1 F 1 = /// F 3 F 2 F 2 = /// F 4 /// F 3 F 3 = /// F 5 /// F 4 F 4 = /// F 6 /// F 5. F n 1 = F////// n+1 /// F n + F n = F n+2 F////// n+1. F 1 + F 2 + F F n = F n+2 1
59 Trying to prove: F 1 + F F n = F n+2 1 We ve just proved a theorem!! Theorem For any positive integer n, the Fibonacci numbers satisfy: F 1 + F 2 + F F n = F n+2 1
60 An example Recall the first several Fibonacci numbers: What is this sum? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, = = 376!!
61 The even Fibonacci Numbers What about the first few Fibonacci numbers with even index: F 2, F 4, F 6,..., F 2n,... Let s call them even Fibonaccis, since their index is even, although the numbers themselves aren t always even!!
62 The even Fibonacci Numbers Some notation: The first even Fibonacci number is F 2 = 2. The second even Fibonacci number is F 4 = 3. The third even Fibonacci number is F 6 = 8. The tenth even Fibonacci number is F 20 =??. The n th even Fibonacci number is F 2n.
63 HOMEWORK Tonight, try to come up with a formula for the sum of the first few even Fibonacci numbers.
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