the fibonacci and lucas sequences
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Definition: The Fibonacci sequence is the sequence (๐น๐)๐โฅ0 with the
recurrence relation ๐น๐โ1 + ๐น๐ = ๐น๐+1 and two initial conditions ๐น0 = 0
and ๐น1 = 1.
Fill in the missing blanks [You Do!]
๐น0 ๐น1 ๐น2 ๐น3 ๐น4 ๐น5 ๐น6 ๐น7 ๐น8 ๐น9 ๐น10 ๐น11 ๐น12 0 1 1 2 3 5
Definition: The Lucas sequence is the sequence (๐ฟ๐)๐โฅ0 with the
recurrence relation ๐ฟ๐โ1 + ๐ฟ๐ = ๐ฟ๐+1 and two initial conditions ๐ฟ0 = 2
and ๐ฟ1 = 1.
Fill in the missing blanks [You Do!]
๐ฟ0 ๐ฟ1 ๐ฟ2 ๐ฟ3 ๐ฟ4 ๐ฟ5 ๐ฟ6 ๐ฟ7 ๐ฟ8 ๐ฟ9 ๐ฟ10 ๐ฟ11 ๐ฟ12 2 1 3 4 7 11
Do you see any connection between these two lists of numbers?
(e.g., add two Fibonacci numbers that are two apart like ๐น4 + ๐น6)
The Fibonacci and Lucas Sequences
FIBONACCI AND LUCAS IDENTITIES
Exercise 1: Add two Fibonacci numbers that are two apart. [You Do!]
๐น0 ๐น1 ๐น2 ๐น3 ๐น4 ๐น5 ๐น6 ๐น7 ๐น8 ๐น9 ๐น10 ๐น11 ๐น12 0 1 1 2 3 5 8 13 21 34 55 89 144
๐ฟ0 ๐ฟ1 ๐ฟ2 ๐ฟ3 ๐ฟ4 ๐ฟ5 ๐ฟ6 ๐ฟ7 ๐ฟ8 ๐ฟ9 ๐ฟ10 ๐ฟ11 ๐ฟ12 2 1 3 4 7 11 18 29 47 76 123 199 322
Conjectured Formula: [You Do!]
Exercise 2: Add Fibonacci and Lucas numbers in same column. [You Do!]
๐น0 ๐น1 ๐น2 ๐น3 ๐น4 ๐น5 ๐น6 ๐น7 ๐น8 ๐น9 ๐น10 ๐น11 ๐น12 0 1 1 2 3 5 8 13 21 34 55 89 144
๐ฟ0 ๐ฟ1 ๐ฟ2 ๐ฟ3 ๐ฟ4 ๐ฟ5 ๐ฟ6 ๐ฟ7 ๐ฟ8 ๐ฟ9 ๐ฟ10 ๐ฟ11 ๐ฟ12 2 1 3 4 7 11 18 29 47 76 123 199 322
Conjectured Formula: [You Do!]
FIBONACCI AND LUCAS IDENTITIES (continued)
Exercise 3: Add two Lucas numbers that are two apart. [You Do!]
๐น0 ๐น1 ๐น2 ๐น3 ๐น4 ๐น5 ๐น6 ๐น7 ๐น8 ๐น9 ๐น10 ๐น11 ๐น12 0 1 1 2 3 5 8 13 21 34 55 89 144
๐ฟ0 ๐ฟ1 ๐ฟ2 ๐ฟ3 ๐ฟ4 ๐ฟ5 ๐ฟ6 ๐ฟ7 ๐ฟ8 ๐ฟ9 ๐ฟ10 ๐ฟ11 ๐ฟ12 2 1 3 4 7 11 18 29 47 76 123 199 322
Conjectured Formula: [You Do!]
Exercise 4: Multiply Fibonacci and Lucas numbers in same column.
[You Do!]
๐น0 ๐น1 ๐น2 ๐น3 ๐น4 ๐น5 ๐น6 ๐น7 ๐น8 ๐น9 ๐น10 ๐น11 ๐น12 0 1 1 2 3 5 8 13 21 34 55 89 144
๐ฟ0 ๐ฟ1 ๐ฟ2 ๐ฟ3 ๐ฟ4 ๐ฟ5 ๐ฟ6 ๐ฟ7 ๐ฟ8 ๐ฟ9 ๐ฟ10 ๐ฟ11 ๐ฟ12 2 1 3 4 7 11 18 29 47 76 123 199 322
Conjectured Formula: [You Do!]
FIBONACCI AND LUCAS IDENTITIES (continued)
Exercise 5: Take the difference of two Fibonacci numbers that are four
apart (e.g., ๐น7 โ ๐น3). [You Do!]
๐น0 ๐น1 ๐น2 ๐น3 ๐น4 ๐น5 ๐น6 ๐น7 ๐น8 ๐น9 ๐น10 ๐น11 ๐น12 0 1 1 2 3 5 8 13 21 34 55 89 144
๐ฟ0 ๐ฟ1 ๐ฟ2 ๐ฟ3 ๐ฟ4 ๐ฟ5 ๐ฟ6 ๐ฟ7 ๐ฟ8 ๐ฟ9 ๐ฟ10 ๐ฟ11 ๐ฟ12 2 1 3 4 7 11 18 29 47 76 123 199 322
Conjectured Formula: [You Do!]
Exercise 6: Conjecture a formula for ๐น๐+1๐ฟ๐+1 โ ๐น๐๐ฟ๐. [You Do!]
๐น0 ๐น1 ๐น2 ๐น3 ๐น4 ๐น5 ๐น6 ๐น7 ๐น8 ๐น9 ๐น10 ๐น11 ๐น12 0 1 1 2 3 5 8 13 21 34 55 89 144
๐ฟ0 ๐ฟ1 ๐ฟ2 ๐ฟ3 ๐ฟ4 ๐ฟ5 ๐ฟ6 ๐ฟ7 ๐ฟ8 ๐ฟ9 ๐ฟ10 ๐ฟ11 ๐ฟ12 2 1 3 4 7 11 18 29 47 76 123 199 322
The Beautiful Binet Formula ๐
EXERCISE 7: What is the 100th Fibonacci number? Can I compute it
without knowing the 98th and 99th Fibonacci numbers? [You Do!]
Hint: Binetโs formula states that
๐น๐ =1
โ5[(
1 + โ5
2)
๐
โ (1 โ โ5
2)
๐
]
EXERCISE 8: What is the 100th Lucas number? Can I compute it without
knowing the 98th and 99th Lucas numbers? [You Do!]
Hint: A โBinetโ-ish formula states that
๐ฟ๐ = (1 + โ5
2)
๐
+ (1 โ โ5
2)
๐
I am Jacques Philippe Marie Binet. I was born in
1786 and died in 1856. The Binet formula is
named after me in 1843, but it was known over
100 years earlier by Abraham de Moivre in 1718.
But my last name is easier to pronounce. Ha.
The Golden Ratio
EXERCISE 9: Where do those two numbers 1+โ5
2 and
1โโ5
2 come from?
Hint: Use the quadratic formula to find the roots of ๐ฅ2 โ ๐ฅ โ 1 = 0.
What does this have to do with beauty?
EXERCISE 10: Let us denote phi (Greek letter ฮฆ) to be this golden. Then
ฮฆ =๐ด
๐ต. Then from the equality
๐ด
๐ต=
๐ด+๐ต
๐ต, we can derive ฮฆ = 1 +
1
ฮฆ.
Multiply this across by ฮฆ and what do you get? [You Do!]
The Golden Ratio (continued)
QUESTION: Does the golden ratio appear everywhere in nature and man?
Here is a picture of some random guy whose hair and side of face model
the golden spiral:
Here is a famous image of the Mona Lisa painting by _______________(?)
The Golden Ratio (continued)
QUESTION: How does the golden ratio and the Fibonacci numbers arise
in sunflowers? [You Do!]
EXERCISE 11: Calculate ๐น๐+1
๐น๐ for ๐ = 9, 10, and 11. What do you notice
about these ratios? Use the Binet formula to calculate ๐น51
๐น50. [You Do!]
The Stair-Climbing Problem
QUESTION: How many ways to climb up two stairs if you can only take
one step or two steps at a time?
ANSWER: [You Draw!]
QUESTION: How many ways to climb up three stairs if you can only
take one step or two steps at a time?
ANSWER: [You Draw!]
QUESTION: How many ways to climb up four stairs if you can only take
one step or two steps at a time?
ANSWER: Here is a picture of the answer.
The answer is 5 ways, since we can do the following five step sequences:
1111, 211, 121, 112, and 22.
The Stair-Climbing Problem (continued)
EXERCISE 12: How many ways to climb up n stairs if you can only take
one step or two steps at a time?
ANSWER: Let ๐ฆ๐ be the number of ways to climb up ๐ stairs if you can
only take one or two steps at a time. For example, we learned on the
previous page that ๐ฆ2 = ______(? ), ๐ฆ3 = ______(? ), and ๐ฆ4 = ______(? ).
So, find a closed form for ๐ฆ_๐. [You Do!]
EXERCISE 13: How many ways to climb up 49 stairs if you can only take
one step or two steps at a time? [You Do!]
The Generalized Fibonacci Sequence
EXERCISE 14: Suppose we start with two initial conditions ๐บ0 = ๐ and
๐บ1 = ๐ and we have the same โFibonacci/Lucasโ recurrence relation
๐บ๐โ1 + ๐บ๐ = ๐บ๐+1 for all ๐ โฅ 1. Prove that we have the following closed
form ๐บ๐ = ๐น๐โ1๐บ0 + ๐น๐๐บ1. [You Do!]
EXERCISE 15: Suppose ๐บ0 = 3 and ๐บ1 = 1. Use the recurrence relation
๐บ๐โ1 + ๐บ๐ = ๐บ๐+1 to compute ๐บ2, ๐บ3, ๐บ4, ๐บ5, and ๐บ6. Verify that ๐บ6
equals 23 by using the closed form ๐บ๐ = ๐น๐โ1๐บ0 + ๐น๐๐บ1. [You Do!]
EXERCISE 16: Suppose ๐บ0 = 7 and ๐บ1 = โ2. Use the recurrence relation
๐บ๐โ1 + ๐บ๐ = ๐บ๐+1 to compute ๐บ2, ๐บ3, ๐บ4, ๐บ5, and ๐บ6. Verify that ๐บ6
equals 19 by using the closed form ๐บ๐ = ๐น๐โ1๐บ0 + ๐น๐๐บ1. [You Do!]
The Fibonacci Matrix
Definition: The Fibonacci matrix is the following 2 ร 2 matrix
๐ = (๐น2 ๐น1
๐น1 ๐น0) = (
1 11 0
).
EXERCISE 17: Do matrix multiplication and compute ๐2, ๐3, ๐4, and ๐5.
What pattern do you notice? Conjecture what matrix you would get if
you had ๐๐. [You Do!]
EXERCISE 18: Compute the determinant (i.e., phattie Dee) for ๐, ๐2, ๐3,
๐4, and ๐5. What pattern do you notice? Conjecture what the
determinant of ๐๐ is. [You Do!]
Pascalโs Triangle โhidesโ the Fibonacci Sequence
EXERCISE 19: Can you find the Fibonacci sequence โhiddenโ inside
Pascalโs triangle below? [You Do!]
Hint: Consider some sums of Pascalโs triangle entries.
EXERCISE 20: Consider the formula below where ๐ = โ๐โ1
2โ is the largest
integer not exceeding ๐โ1
2. For example, if ๐ = 8, then โ
8โ1
2โ = 3 since
7
2= 3
1
2 . Verify the formula below for ๐ = 8. [You Do!]
Pascalโs Triangle โhidesโ the Fibonacci Sequence
(continued)
EXERCISE 21: Consider the 4th row of Pascalโs triangle. Multiply each
number left-to-right by the Fibonacci numbers ๐น0, ๐น1, ๐น2, etc. Then add
the numbers in that row. What do you notice? [You Do!] Do the same for
the 5th row. What do you notice? [You Do!]
EXERCISE 22: Conjecture a closed form for the following sum in the ๐๐กโ
row of Pascalโs triangle: [You Do!]
(๐
0) ๐น0 + (
๐
1) ๐น1 + (
๐
2) ๐น2 + โฏ + (
๐
๐ โ 1) ๐น๐โ1 + (
๐
๐) ๐น๐.
Back Page History
Who is Fibonacci?
โข He lived from around 1170 to 1240. He was born in the Republic of
Pisa (currently Italy).
โข His name is NOT Fibonacci. The term is short for the Latin โfilius
Bonnaciโ (i.e., the son of the Bonnacioโs; his fatherโs name was
Guglielmo Bonnacio).
Who is รdouard Lucas?
โข He lived from 1842 to 1891. He was born in France.
โข He died in unusual circumstances. At a banquet, a waiter dropped
some crockery and a piece of broken plate cut Lucas on the cheek.
He died a few days later of a severe skin inflammation probably
caused by sepsis. He was only 49 years old.
Is This Page Intentionally Left Blank for Your Doodling Delight? Nope. It is placed here to ensure that the new section on the next page
begins with an odd-numbered page.
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