CHAPTER 1
Nature of Mathematics
Rationale
Learning Objectives:
1. Know the theories and concepts of nature of mathematics
2. Be able to identify different formulas In patterns and sequence
3. To familiarize with the symbols, formula and establish rules in each topic
Learning Outcomes:
1. Develop the mathematical analysis to understand sequence and patterns.
2. Recognize patterns and appreciate the Nature of Mathematics in the Modern World
PATTERN
A pattern is a series or sequence that repeats. You can observe patterns - things like colors,
shapes, actions, or other sequences that repeat - everywhere. Think about words or melodies in
songs, lines and curves on buildings, or even in the grocery store where boxes and jars of various
items are lined up.
But, one of the most common places to find patterns is in math. Math patterns are sequences
that repeat according to a rule or rules.
Example:
1. Find the pattern. 1, 4, 7, 10, 13, 16, _?
Answer: starts at 1 and jumps 3 every time.
19
2. Find the pattern.
The rectangular numbers are the next sequence in the progression of patterns. Instead of
the expression
(or
) as with square numbers, the rectangular numbers use
for one dimension.
EXPONENTIAL GROWTH MODEL
In exponential growth, a population's per capita (per individual) growth rate stays the same
regardless of population size, making the population grow faster and faster as it gets larger.
In nature, populations may grow exponentially for some period, but they will ultimately be
limited by resource availability.
The formula is A=Pert
where:
A= the size of the population after it grows
P= the initial number of people
r= rate of growth
t= time
e= 2.718
Example:
A=1,236,000; r=12%, i=50 years, P=?
Find the population number.
Answer: Since the formula is A=Pert , transposition would be an appropriate step to get the initial
number of people
P=Aert
P= 1,236,000 (2.718(12%)(50))
P=1,236,000 (2.718(6))
P=1,236,000 (403.1778962)
P=498,327,880
Sequence
A list of numbers or objects in a special order. It is a particular order in which related things
follow each other.
Example:
3, 5, 7, 9, ... is a sequence starting at 3 and increasing by 2 each time.
Fibonacci Sequence
The Fibonacci sequence is a set of numbers that starts with a one or a zero, followed by a one,
and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum
of the preceding two numbers. If the Fibonacci sequence is denoted F (n), where n is the first
term in the sequence, the following equation obtains for n = 0, where the first two terms are
defined as 0 and 1 by convention.
Example:
F (0) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ...
Patterns
1.
2.
2. What letter comes next in this pattern?
ZEF YGH XIJ
WKL _____
3. What completes the following pattern?
50, 55, 65, 80, 100, ___
Fibonacci sequence
Find F(39).
1
1+√5
1−√5
2
2
F(39) = √5 [(
)]39 − [(
)]39
F(39) =63,245,986
If Fib(18) = 2,854 and Fib (19) = 4, 181
What is Fib (20) and Fib (21)
F(20) =
1
1+√5
1−√5
2
2
[(
√5
F(20) = 6,765
)]20 − [(
)]20
1
F(21) = √5 [(
1+√5
1−√5
2
2
F(21) = 10,946
)]21 − [(
)]21
Evaluate the following:
a. Fib(31) + Fib(32) =
b. Fib (15) + Fib(16) +Fib(17)=
Suppose the population in a city is 65,000. The population size is given by the exponential
growth model A= 65,000𝑒 0.65 . Find t when the population would be