1. By:
Arlene M. Leron

2. Objectives:
At the end of the lesson, you should be
able to perform the following tasks:
1. Define fibonacci sequence and give its
significance.
2. Find the next terms of the given fibonacci
sequence.
3. Investigate the relationship between the
Fibonacci sequence with the golden ratio.

3. Who Was Fibonacci?
• European
Mathematician 1175-
1250
• Real name Leonardo
of Pisa.
• Author of Liber Abbaci
or Book of the Abacus
or Book of Calculation

4. What Is the Fibonacci Sequence and
Why Is It Significant?
• Generalized
sequence of first
two positive
integers and the
next number is
the sum of the
previous two, i.e.
1,1,2,3,5,8,13,21
,…

5. What Is the Fibonacci Sequence and
Why Is It Significant?
• Shows up
unexpectedly in
architecture,
science and
nature
(sunflowers &
pineapples).

6. What Is the Fibonacci Sequence and
Why Is It Significant?
• Has useful
applications with
computer
programming,
sorting of data,
generation of
random numbers,
etc.

9. Fibonacci Sequence
• The Fibonacci Sequence 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.
• 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),
• And the 5 is (2+3), and so on!

10. Makes a Spiral
When we make squares with those
widths, we get a nice spiral:

11. The Rule
xn = xn-1 + xn-2
where:
xn is term number "n"
xn-1 is the previous term (n-1)
xn-2 is the term before that (n-2)

12. n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 ...
Example: the 8th term is
the 7th term plus
the 6th term:
x8 = x7 + x6
First, the terms are numbered from 0 onwards
like this: So term number 6 is called x6 (which
equals 8).

13. n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ...
 Look at the number x3 = 2. Every 3rd number is a multiple
of 2 (2, 8, 34, 144, 610, ...)
 Look at the number x4 = 3. Every 4th number is a multiple
of 3 (3, 21, 144, ...)
 Look at the number x5 = 5. Every 5th number is a multiple
of 5 (5, 55, 610, ...)

14. Golden Ratio
• In mathematics, two quantities are in the Golden
ratio if their ratio is the same of their sum to the
larger of the two quantities.
• “De Devina Proportione” by Luca Paciolli

15. Golden Ratio
• Is the relationship between numbers on the
Fibonacci sequence where plotting the
relationships between on scales results in a
spiral shape.

16. Golden Ratio

17. Golden Ratio

18. Golden Ratio

19. Seatwork
If the first three fibonacci
numbers are given as
1,1,2,…, then what is the
least value of n for which Xn
> 500? Show your
solutions.