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1. MATHEMATICS IN THE MODERN WORLD
A. Y. 2025 – 2026
NATURE OF MATHEMATICS

2. PROPERTIES
01
PATTERNS AND NUMBERS IN NATURE
AND THE WORLD
02
THE FIBONACCI SEQUENCE
03
MATHEMATICS FOR OUR WORLD

3. SEQUENCE
It is an ordered list of numbers, called terms, that may
have repeated values. The arrangement of these
terms is set by a definite rule.

4. FIBONACCI SEQUENCE
LEONARDO PISANO BOGOLLO BREEDING AND REPRODUCTION OF RABBITS

5. FIBONACCI SEQUENCE
Discovery of this sequence:
Breeding and Reproduction of
Rabbits
LEONARDO PISANO BOGOLLO
He noted that the set of
numbers was generated by
getting the sum of the two
previous terms.

6. FIBONACCI SEQUENCE
Starting with 0 and 1, the succeeding terms in
the sequence can be generated by adding two
numbers that came before the term:
0, 1, 1, 2, 3, 5, 8,13,21, 34, 55, 89,
Fn = Fn – 1 + Fn - 2

7. EXAMPLE:
Let 𝐹𝑛 be the nth terms of the Fibonacci Sequence,
with 𝐹1 = 1, 𝐹2 = 1 , 𝐹3 = 2, and so on.
A. Find F7
Fn = Fn – 1 + Fn - 2
F7 = F7 – 1 + F7 - 2
F7 = F6 + F5
F7 = 8 + 5
F7 = 13

8. EXAMPLE:
Let 𝐹𝑛 be the nth terms of the Fibonacci Sequence,
with 𝐹1 = 1, 𝐹2 = 1 , 𝐹3 = 2, and so on.
B. Find F12
F12 = F12 – 1 + F12 - 2
Fn = Fn – 1 + Fn - 2
F12 = F11 + F10
F12 = 89 + 55
F12 = 144

9. EXAMPLE:
Let 𝐹𝑛 be the nth terms of the Fibonacci Sequence,
with 𝐹1 = 1, 𝐹2 = 1 , 𝐹3 = 2, and so on.
B. Find F20
Fn = Fn – 1 + Fn - 2
F20 = F20 – 1 + F20 - 2
F20 = F19 + F18
F20 = 4181 + 2584
F20 = 6765

10. EXAMPLE:
Let 𝐹𝑛 be the nth terms of the Fibonacci Sequence,
with 𝐹1 = 1, 𝐹2 = 1 , 𝐹3 = 2, and so on.
D. Evaluate: F1 + F2 + F3 + F4 + F5 = _____.
1 + 1 + 2 + 3 + 5 = 12

11. What is the 80th term of the Fibonacci
Sequence?

12. BINET’S FORMULA
It may be used to find the nth term
of a Fibonacci Sequence using
Binet’s Formula.
JACQUES PHILIPPE
MARIE BINET

13. EXAMPLE:
Let 𝐹𝑛 be the nth terms of the Fibonacci Sequence,
with 𝐹1 = 1, 𝐹2 = 1 , 𝐹3 = 2, and so on.
B. Find F20
F20 = 6765

14. What is the 80th term of the Fibonacci Sequence?

15. The Fibonacci sequence has many
interesting properties. Among these is
that this pattern is very visible in nature.

16. It is also interesting to note that the ratios of
successive Fibonacci numbers approach the
number (Phi), also known as the Golden
Ratio. This is approximately equal to 1.618.

17. THE GOLDEN RATIO
It can also be expressed as the ratio between two
numbers if the latter is also the ratio between the
sum and the larger of the two numbers.
𝜙 ≈ 1.618

18. 1
1
= 1.0000
2
1
= 2.0000
3
2
= 1.5000
5
3
≈ 1.6667
8
5
= 1.6000
13
8
= 1.6250
21
13
≈ 1.6154
34
21
= 1.6190
55
34
≈ 1.6177
89
55
≈ 1.6182
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …
144
89
≈ 1.6180
233
144
≈ 1.6180

19. FIBONACCI SPIRAL
Geometrically, it can also be visualized as a rectangle perfectly formed by a
square and another rectangle, which can be repeated infinitely inside each
section.

20. FIBONACCI SPIRAL
Shapes and figures that bear this
proportion are generally considered to be
aesthetically pleasing.

21. MONA LISA NOTRE DAME CATHEDRAL

22. PARTHENON

23. PROPERTIES
01
PATTERNS AND NUMBERS IN NATURE
AND THE WORLD
02
THE FIBONACCI SEQUENCE
03
MATHEMATICS FOR OUR WORLD

24. MATHEMATICS FOR ORGANIZATION
Mathematics helps organize
patterns and regularities in the
world.

25. MATHEMATICS FOR PREDICTION
Mathematics helps
predict the behavior of
nature and phenomena in
the world.

26. MATHEMATICS FOR CONTROL
Mathematics helps control
nature and occurrences in
the world for our own ends.

27. MATHEMATICS IS INDISPENSABLE
Mathematics has
numerous applications
in the world making it
indispensable.