This document discusses patterns and sequences in mathematics. It provides examples of different types of patterns and sequences found in nature as well as number patterns like arithmetic, geometric, and Fibonacci sequences. Formulas are given for exponential growth, arithmetic sequences, geometric sequences, and the Fibonacci sequence, along with examples of using the formulas to analyze patterns and sequences.

1. Mathematics
In the
Modern
World
(Nature of
Mathematics
JASMIN C. TAWANTAWAN, LPT, Ph.D (CAR)

2. ◍ Pattern – are
regular,
repeated, or
recurring
forms or
designs.
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4. Nature forms of
pattern
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5. Roadmap
5
1 3 5
6
4
2
Snowflakes and
Honeycombs
Tigers’ stripe and
hyenas’ Spots
Snail’s shell
Order of
rotation
Sunflower Flower petals
5
World
population

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13. “
Formula of exponential growth:
𝑨 = 𝑷𝒆𝒓𝒕
Where,
A - is the size of the population
after in grows
P - is the initial number of people
r - is the rate of growth, and
t - is time
e – is Euler’s constant ≈ 𝟐. 𝟕𝟏𝟖
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14. This is a slide title
◍ Example:
The exponential growth model 𝐴 =
30𝑒0.02𝑡
describes the population of a
city in the Philippines in thousands, t
year after 1995.
a. What is the population of the city
in 1995?
b. What will be the population in 2017?
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15. ◍ Sequence – is an
order list of
numbers, called
terms, that may have
repeated values. The
arrangement of these
terms is set by
definite rule.
Sequence
Example: Analyze
the given sequence
for its rule and
identify the next
three terms.
a. 1, 10, 100, 1000
b. 2,5,9,14, 20
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16. Types of Number Patterns in Math
1. Arithmetic sequence
◍ An arithmetic sequence is
a sequence where every
term after the first is
obtained by adding a
constant called the
common difference.
In general the nth term of
a given sequence:
◍ 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏
◍ 𝑺 =
𝒏
𝟐
(𝒂𝟏 + 𝒂𝒏)
◍ Example:
1. What is the 12th term
of the arithmetic
sequence 0, 5, 10, 15,
20, 25,…?
2. Find the sum of
arithmetic sequence 0,
5, 10, 15, 20, 25.
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17. 2. Geometric Sequence
◍ A geometric sequence is a
sequence where each term
after the first is
obtained by multiplying
the preceding term by a
nonzero constant called
the common ratio.
◍ 𝒂𝒏 = 𝒂𝟏
𝒓𝒏−𝟏.
◍ Example. Find
the common ratio
of the sequence
32, 16, 8, 4, 2,
... .
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18. 3. Fibonacci Sequence
The Fibonacci sequence
is defined by the
recursive formula
𝑭𝒏
= 𝑭𝒏−𝟐 + 𝑭𝒏−𝟏, 𝒘𝒉𝒆𝒓𝒆 𝑭𝟏
= 𝑭𝟐 = 𝟏
Example 1. Given the
recursive formula for the
Fibonacci sequence
𝐹𝑛 = 𝐹𝑛−2 + 𝐹𝑛−1,
𝑤ℎ𝑒𝑟𝑒 𝐹1 = 𝐹2 = 1.
a. 𝐹3
b. 𝐹4
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19. Mathematics for our world
◍ Mathematics
for
Organization
Ex. Sales,
internet, social
media, growth,
ideas, data, &
etc.
◍ Mathematics for
Prediction
Ex. Applying
concept of
probability,
historical
pattern,
metrological,
weather, & etc.
◍ Mathematics
for Control
Ex.
Gravitational
waves, threat
of climate
change
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20. Thanks!
Any questions?
👍
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21. 21
Activity 1
Determine what comes next in the given pattern.
1. A, C, E, G, I, ______
2. 15, 10, 14, 10, 13, 10, ____
3. 3,6,12,24,48,96,____
4. 27,30,33,36,39,_____
5. 41,30,37,35,33,____
Find the missing quantity.
1. 𝑃 = 680,000; 𝑟 = 12% 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟; 𝑡 = 8 𝑦𝑒𝑎𝑟𝑠
2. 𝐴 = 1,240,000; 𝑟 = 8% 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟; 𝑡 = 30 𝑦𝑒𝑎𝑟𝑠