CHAPTER 1: MATHEMATICS IN OUR WORLD
1.3 THE FIBONACCI SEQUENCE AND THE GOLDEN RATIO
Learning Outcomes:
At the end of this module, the learners are expected to:
-
Identify patterns in nature and regularities in the world.
Determine the roles and applications of mathematics both in nature and in human
endeavors.
Express appreciation for mathematics.
I. Introduction
The Fibonnaci numbers is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34,...
The next number is found by adding up the two numbers before it:
For instance,
-
The 2 is found by adding the two numbers before it (1+1)
The 3 is found by adding the two numbers before it (1+2)
The 5 is found by adding the two numbers before it (2+3)
and so on..
GET TO KNOW
Leonardo of Pisa (c. 1170- c. 1250)
-
Fibonacci is named after Leonardo of Pisa,
also known as Fibonacci.
Fibonacci’s 1202 book Liber Abaci introduced
the sequence to Western European
mathematics, although it had been described
earlier in Indian Mathematics.
The Rule
The Fibonacci sequence can be written as a “Rule”
As observed, the terms are numbered from 0 onwards like this:
n=
0
1
2
3
4
5
6
7
8
9
10
11
12
13
𝐹𝑛 =
0
1
1
2
3
5
8
13
21
34
55
89
144 233 377
In mathematical terms, the sequence 𝐹𝑛 of Fibonacci numbers is defined by the
recurrence relation
𝑭𝒏 = 𝑭𝒏−𝟏 + 𝑭𝒏−𝟐
Where:
-
𝐹𝑛 - is term number “n”
𝐹𝑛−1 – is the previous term (n-1)
𝐹𝑛−2 – is the term before than (n-2)
Example 1: The term 10 is calculated like this:
𝐹10 = 𝐹10−1 + 𝐹10−2
𝐹10 = 𝐹9 + 𝐹8
𝐹10 = 34 + 21
𝐹10 = 55
14
To find the 𝑛𝑡ℎ Fibonacci number without using the recursion formula, the following is
evaluated using a calculator:
𝑛
𝐹𝑛 =
𝑛
1 + √5
1 − √5
( 2 ) −( 2 )
√5
This form is known as the Binet form of the 𝒏𝒕𝒉 Fibonacci number.
Example 2: Use Binet’s formula to determine the 25th Fibonacci number.
𝑛
𝐹𝑛 =
𝐹25 =
𝑛
1 + √5
1 − √5
( 2 ) −( 2 )
√5
1 + √5
( 2 )
25
25
1 − √5
−(
2 )
√5
𝐹𝑛 = 75,025
The Fibonacci numbers can also be used to define a spiral. Fibonacci numbers can be
illustrated as a spiral with squares representing the widths of the numbers in sequence.
©Oleksii Arseniuk/Getty Images
Figure 1: Fibonacci numbers spiral
Shown in Figure 1 is the Fibonacci numbers spiral. The squares fit nicely together,
forming the spiral. For instance, 5 plus 8 equals 13, then 8 plus 13 add up to 21, etc.
Fibonacci numbers are of interest to biologist and physicist, artists, and designers for
centuries because they are usually observed in various natural objects and phenomena.
For instance,
-
The branching patterns in trees and leaves
The distribution of seeds in a raspberry
Spiral galaxies
Shells
Faces
Flower petals
Hurricanes
CHECK YOUR
UNDERSTANDING
__________1. The Fibonacci sequence looks like this:
a. 1,1,2,5,9,...
b. 0, 1, 2, 3, 4, 5, 6, 7, ...
c. 0, 1, 1, 2, 3, 5, 8, 13, ...
d. 1, 2, 4, 6, 8, ...
___________2. The Fibonacci sequence begins with what two numbers?
a. 1 and 1
b. 0 and 1
c. 1 and 2
d. 2 and 3
3. The first four numbers in the Fibonacci sequence are 1,1,2,3,...
(a) What is the 5th term of the Fibonacci sequence?
(b) What is the 6th term of the Fibonacci sequence?
4. Beth wants to generate a Fibonacci style sequence. The first term is 3 and the
second term is 4.
(a) Find the 3rd term.
(b) Find the 4th term.
5. Find the nth Fibonacci using the Binet formula. Write your solution.
(a) n = 14
(b) n = 20