1.1 MATHEMATICS IN OUR WORLD 1.

Patterns can be observed even in stars

that move in circles across the sky each
WHAT IS MATHEMATICS? day.
2.  The weather season cycle each year.
 Mathematics is the study of pattern and All snowflakes contain six-fold symmetry
structure. which no two are exactly the same.
 Mathematics is fundamental to the
physical and biological sciences, 3. Patterns can be seen in fish patterns
engineering and information technology, to like spotted trunkfish, spotted puffer, blue
economics, and increasingly to the social spotted stingray, spotted moral eel, coral
sciences. grouper, red lionfish, yellow boxfish, and
 Mathematics is a useful way to think angelfish. These animals and fish stripes
about nature and our world. and spots attest to mathematical
 Mathematics is a tool to quantify, regularities in biological growth and form.
organize, and control our world, predict
phenomena, and make life easier for us. 4. Zebras, tigers, cats, and snakes are
covered in patterns of stripes; leopards and
WHERE IS MATHEMATICS? hyenas are covered in a pattern of spots,
and giraffes are covered in a pattern of
 Many patterns and occurrences exist in blotches.
nature, in our world, in our life.
 Mathematics helps make sense of these 5. Natural patterns like the intricate waves
patterns and occurrences. across the oceans; sand dunes on deserts;
formation of the typhoon; water drop with
WHAT ROLE DOES MATHEMATICS PLAY ripple and others. These serve as clues to
IN OUR WORLD? the rules that govern the flow of water,
sand, and air.
 Mathematics helps organize patterns
and regularities in our world. 6. Other patterns in nature can also be
 Mathematics helps predict the behavior seen in the ball of mackerel, the v-formation
of nature and phenomena in the world. of geese in the sky, and the tornado
 Mathematics helps control nature and formation of starlings.
occurrences in the world for our own ends.
 Mathematics has numerous applications
in the world making it indispensable. PATTERNS AND REGULARITIES

PATTERNS AND NUMBERS IN NATURE Mathematics is all around us. As we discover

AND THE WORLD more about our environment, we can
mathematically describe nature.
 Patterns in nature are visible regularities The beauty of a flower, the majestic tree, and
of form found in the natural world and can even the rock formation exhibits nature’s sense
also be seen in the universe. of symmetry.
 Nature patterns which are not just to be
admired, they are vital clues to the rules Have you ever thought about how nature likes
that govern natural processes. to arrange itself in patterns in order to act
efficiently?
Check out some examples of these patterns Nothing in nature happens without a reason, all
that you may be able to spot the moment of these patterns have an important reason to
you decided to go for a walk. exist.
 3. SPIRALS - are a curved pattern that
focuses on a center point and a series of
circular shapes that revolve around it.
TYPES OF PATTERNS
1. SYMMETRY – a pattern with a sense of A logarithmic spiral or growth spiral is a self-
harmonious and beautiful proportion of balance similar spiral curve which often appears in
or an object is invariant to any various nature.
transformations. Examples are reflection, It was first described by Rene Descartes and
rotation or scaling. was later investigated by Jacob Bernoulli. 
1.1 Bilateral Symmetry - is symmetry Examples of spirals are pine cones,
in which the left and right sides of the organism pineapples, hurricanes.
can be divided into approximately mirror
images of each other along the midline. This 1.2 GOLDEN RATIO
exists in living things like insects, animals,
plants, flowers, and others. GOLDEN RATIO

Animals can further be classified as  It was first called the Divine

either cyclic or dihedral. Proportion in the early 1500s in Leonardo
da Vinci’s work.
Plants on the other hand often have radial or
rotational symmetry, as to flowers and some
 Mathematically two quantities are in the
group of animals.
Golden ratio if (a+b) divided by a is equal to
There is also what we call a five-fold symmetry a divided by b which is equal to
which is found in the echinoderms, the group 1.618033987…and represented by (phi),
which includes starfish (dihedral-D5 provided that a is greater than b.
symmetry), sea urchins, and sea lilies.
Examples:
Radial symmetry suits organisms like sea
anemones whose adults do not move and 1. If a = 3 and b = 2 then a/b = 1.5
jellyfish (dihedral-D4 symmetry). Radial 2. If a = 5 and b = 3 then a/b = 1.666666...
symmetry is also evident in different kinds of 3. If a = 8 and b = 5 then a/b = 1.6
flowers. 4. if a = 13 and b = 8 then a/b= 1.625
5. If a = 21 and b = 13 then a/b =
2. FRACTALS – a pattern with a curve or 1.615384615...
geometric figure, each part of which has the
same statistical character as the whole.    These examples show the relationship
between a and b which represents a golden
A fractal is a never-ending pattern found in ratio.
nature. The exact same shape is replicated in
a process called “self- similarity.”    The quotient of a and b is somewhat close to
the value of a golden ratio which happens to
The pattern repeats itself over and over again be equal to 1.618033987...
at different scales. For example, a tree grows
by repetitive branching.     In the same manner, the golden ratio can
also be noticed in Arts let us name a few...
This same kind of branching can be seen in
lightning bolts and the veins in your body. 1. The exterior dimension of the Pathernon
Now, try to examine a single fern or an aerial in Athens, Greece embodies the golden
view of an entire river system, find out if you’ll ratio.
see fractal patterns. 2. In Timaeus Plato describes five
possible regular solids that relate to the
 golden ratio which is now known  The mathematical language is the
as Platonic Solid. system used to communicate mathematical
3. Euclid was the first to give a definition ideas.
of the golden ratio as a dividing line in the  This language consists of some natural
extreme and mean ratio in his book language using technical terms
the Elements. (mathematical terms) and grammatical
4.  Leonardo da Vinci used the golden conventions that are uncommon to
ratio to define the fundamental portions of mathematical discourse, supplemented by
his works. He incorporated the golden ratio a highly specialized symbolic notation for
in his paintings such as" The Last Supper", mathematical formulas.
"Monalisa" and "St. Jerome in the
Wilderness". - The mathematical notation used for formulas
5.  Michael Angelo di Lodovico has its grammar and shared by
Simon was considered the greatest living mathematicians anywhere in the globe.
artist of his time. He used the golden ratio
in his painting " The Creation of Adam ".  Mathematical language must be precise,
6. Raffaello Sanzio da Urbino was a concise, and powerful, these must be its
painter and an architect from a characteristics.
renaissance. In his painting "The School of
Athens", the division between the figures in 1.3 MATHEMATICAL LANGUAGE AND
the painting and their proportions are SYMBOLS
distributed using the golden ratio.
CHARACTERISTICS OF MATHEMATICAL
Golden Ratio in Architecture: LANGUAGE
Let us have some architectural structures that 1.) Precision in mathematics is a culture of
exhibit the application of the Golden Ratio: being correct all the time. Definition and limits
should be distinct. Mathematical ideas are
1.  Great Pyramid of Giza built 4700 BC in being developed informally and being done
Ahmes Papyrus of Egypt, the ratio of its more formally, with necessary and sufficient
base to the height is roughly 1.5717 which conditions stated upfront and restricting the
is close to the golden ratio. discussion to a particular class of objects.
2. Notre Dame is a Gothic Cathedral in
Paris. 2.) Concise in mathematics must show
3. Taj Mahal is found in India and used the simplicity. Being concise is a strong part of the
golden ratio in its construction and was culture in mathematical language.
completed in 1648. Mathematicians desire the simplest possible
4. Cathedral of Our Lady of Chartres in single exposition. 
Paris, France which also exhibits the
3.) Mathematical language must also
golden ratio.
be powerful. It is a way of expressing complex
5. The United Nation Building, the
thoughts with relative ease. The abstraction in
window configuration reveals the golden
mathematics is the desire to unify diverse
proportion.
instances under a single conceptual framework
6. Eiffel Tower, found in Paris France, and
and allows easier penetration of the subject
erected in 1889 which is an iron lattice.
and the development of more powerful
7. CN Tower in Toronto, the tallest tower,
methods.
and free-standing structure in the world,
contains the golden ratio in its design. How does Expression differ from sentences?

MATHEMATICAL LANGUAGE EXPRESSION VERSUS SENTENCES

 An expression (or mathematical Ex. Axiom, conjecture, theorems, lemma, and
expression) is a finite combination of corollaries.
symbols that is well-defined according to
rules that depend on the context. The 3. Mathematics also has Mathematical
symbols can designate numbers, variables, jargon- mathematical phrases used with
operations, functions, brackets, specific meanings.
punctuation, and groupings to help
determine the order of operations and other Ex. “If and only if”, “necessary and sufficient”
aspects of mathematical syntax. and “without loss of generality.”
 An expression is a correct arrangement
of mathematical symbols used to represent 4. The vocabulary of mathematics also has
the object of interest, it does not contain a visual elements.
complete thought, and it cannot determine
if it is true or false. Some types of Ex. Used informally in blackboards and
expressions are numbers, sets, and formally in books and researches which serve
functions. to display schematic information so easily.
 A sentence (or mathematical sentence)
makes a statement about two expressions, 5. The mathematical notation has its
either using numbers, variables, or a grammar and does not dependent on a
combination of both. A mathematical specific natural language.
sentence can also use symbols or words
like equals, greater than, or less than. Ex. Latin alphabet used for simple variables
 A mathematical sentence is a correct and parameters.
arrangement of mathematical symbols that
states a complete thought and can be 6. Mathematical expressions containing a
determined whether it’s true, false, and symbolic verb are generally treated as
sometimes true/sometimes false. clauses in sentences or as a complete
sentence and are punctuated as such by
CONVENTIONS IN THE MATHEMATICAL mathematicians.
LANGUAGE
Ex. Equal ( = ) , Less than ( < ) , Greater than ( > ) ,
Mathematical languages have conventions and Addition (+), Subtraction (-) , Multiplication (x),
it helps individuals distinguish between infinity ( ∞), for all ( ∀) , there exists (∋ ), element
different types of mathematical expressions. (∈ ) , implies (⟶ ),if and only if (⟷ ), therefore ( ∴)
etc.
The mathematical convention is a fact, name,
notation, or usage which is generally agreed 1.4 FOUR BASIC CONCEPTS IN MATHEMATICS
upon by mathematicians. (1.5.1 LANGUAGE OF SET)

Mathematicians abide by conventions to be Set Theory is the branch of mathematics that

able to understand what they write without studies set or the mathematical science of the
constantly having to redefine basic terms. infinite.
1. Mathematics has its brand of technical  The study of sets has become a
terms. – a word in general usage has a fundamental theory in 1870.
different and specific meaning within
 Introduced by George Cantor (German
mathematics.
Mathematician.)
Ex. Group, ring, field, term, factor, etc.
SET
2. Mathematical statements also have their
 is a collection of well-defined objects.
taxonomy.
 usually denoted by capital letters of the         D= {x/x are vowel letters from the
alphabet and its members are enclosed alphabet}
with brackets.
2. E= {4, 6, 8, 10, 12, 14, 18, 20}
Elements – are the members or objects of the
set which is denoted by a symbol (∈ ).          E= {x/x is even number from 4 to 20}
Example of a set: 3. F= {12}
        A-{ x/x is a set of letters from the word
Pneumonia}          F= [x/x is equal to 12}

         This is read as A is the set of all x Cardinal Number- this refers to the number of
such that x is a set of letters from the word elements in a given set. The cardinality of a set
Pneumonia. is given by n(A).

        The elements of this set are a, e, i, m, Examples: Identify the Cardinality of the given
n, o, p, u. sets.

TWO WAYS OF REPRESENTING A SET 1. A= {1, 2, 3, 4, 5, 6, 7, 8, 9} 

1. ) Roster Method (Tabulation          n(A)=9

Method) – when the elements of the set
are enumerated and separated by a 2. B= {x/x is a month in the calendar}
comma.
        n(B)=12
             Ex.    A= {23, 25, 27}
3. C= {x/x is an integer, 1< x < 8}
Write the following sets in Roster Method:
     n(C)= 6
1. A= {x/x is a positive integer less than
10}  4. D= {a, e, i, o, u}

         A= {1, 2, 3, 4, 5, 6, 7, 8, 9}        n(D)= 5

2. B= {x/x is a month in the calendar} 5. E= {4, 6, 8, 10, 12, 14, 18, 20}

        B= {Jan, Feb, Mar, Apr, May, Jun, Jul,       n(E) = 8

Aug, Sept, Oct, Nov, Dec}
6. F= {12}
3. C= {x/x is an integer, 1< x < 8}
      n(F) = 1
        C= {2, 3, 4, 5, 6, 7} TYPES OF SET
2.) Rule Method (Set builder notation) - used
to describe the elements or members of the set 1. Finite Set – is a set whose elements
using their common characteristics. are limited or countable and the last
element can be identified.
       Ex.   B= {x/x is a set of professors from 2. Infinite Set – is a set whose elements
the Math and Physics Department} are unlimited or uncountable and the last
element cannot be specified.
Write the following Set in Rule Method 3. Unit Set – is a set with only one
element, it is also called
1. D= {a, e, i, o, u}
4. Empty Set – a unique set with no            F = { 1 , 5 }
elements and also called as the Null Set. It
is denoted by { }.            U = { a , b , c, d, e, f, g, h,  i, j, 1, 2, 3,
5. Universal Set – the totality of the set, all 4, 5}
sets under investigation in any application Based on the definitions of each set we can
of set theory are assumed to be contained have the following:
in some largely fixed set and is denoted
by U.
6. 6. Subset - if A and B are set, A is  B is a subset of C, mathematically 
called a subset of B, written A ⊆ B, if and B ⊆ CB ⊆ CB ⊆ C
only if, every element of A is also an  A is a proper subset of B,
element of B. A is a proper subset of B, mathematically A ⊂ BA ⊂ BA ⊂ B
written A ⊂ B, if and only if, every element  B and C are equal set, they have the
of A is in B but there is at least one element same elements.
of B that is not in A.  D is equivalent to E,
7. Equal Set - two sets are equal if and
only if, every element of A is in B and every mathematically  D ∼ ED ∼ ED ∼ E,
element of B is in A. these sets have the same number of
8. Equivalent Set - two sets are elements.
equivalent if they have the same number of  B is equivalent to C,
elements and it is denoted by (~).
9. 9. Disjoint set - two sets that do not mathematically  B ∼ CB ∼ CB ∼ C,
have the same elements. This is also these sets have the same number of
known as a non-intersecting set. elements.
 D and E are disjoint sets.
Examples: Classify the given sets as a finite
set, infinite set, unit set, empty set, and 1.4.1 SET OPERATIONS
universal set.
OPERATIONS ON SETS:
Given:
1. UNION OF SET- the union of A and B,
1. A = { a, b , c } = finite set
denoted by A ∪ B, is the set of all elements
2. B = { a, b , c, d, e } = finite set
in x in U such that x is in A or x is in B.
3. C = { a, b , c, e, d.... } =infinite set
2. INTERSECTION OF SET - the
4. D = { }  = empty set/ null set
intersection of A and B, denoted by A ∩ B,
5. E = { bat } = unit sets
is the set of all elements in x in U such that
6. U = { a , b , c, d, e, bat } = universal
x is in A and x is in B.
set
        Given:  A = { a, b, c }
Examples: Which of these are subsets,
equal sets, equivalent sets, and disjoint                         B = { c, d, e }
set.    
                        C = { f, g }
Given: 
                        D = { f, g, h, i}
         A = { a , b , c }
Let us answer the set of examples:
          B = { a , b , c, d, e }
a. A ∪ B = { a , b, c, d, e }             
          C = { a , b , c, e, d } b. C ∪ D = { f , g, h, i }   
c. B ∪ C = { c, d, e, f, g }
           D = { f , g, h , i } d. A∩B={c}
e. C ∩ D = { f, g }
           E = { 1, 2, 3, 4 }
f. B ∩ C = {   }
3. COMPLEMENT OF SET- The complement 1.4.2 VENN DIAGRAM
of a set or absolute complement A, denoted by
A' , is the set of all elements in x in U such that Venn Diagram- is an illustration that uses
x is not in A. circles to show the relationships among things
or finite groups of things.
 Given:  A= { a, b, c }
 Circles that overlap have a commonality
              B= { c, d, e } while circles that do not overlap do not
              U = { a, b, c, d, e, f, g, h } share those traits.
 The circles are being placed inside a
 Find the following: box, where the box represents the universal
set and the shaded inside of a circle
a. A'   = { d, e, f, g, h}     represents the subset of a universal set.
b. B'   = { a , b, f, g, h }        
c. ( A ∩ B ) ′ = { a , b , d , e , f , g , h }
Sometimes we will use the Venn Diagram for a
( A ∩ B ) ′ = { a , b , d , e , f , g , h }
d. ( A' ∩ B' ) ={ f, g, h }
particular set whose elements are known, the
elements should be listed accordingly.
4. DIFFERENCE OF SET - The difference of A Given: U = { 1, 2, 3, 4, 5 } and A = { 2 , 4 }
and B (or relative complement of B with
respect to A) , denoted by  A - B, is the set of  Let us illustrate the representation of these
all elements x in U such that x is in A and x is sets in the Venn diagram.
not in B. 

         Given:  A= { a, b, c }
Disjoint Set can also be represented in a Venn
                     B = { c, d, e } Diagram:
                     C = { f, g }
 
                     D = { f, g, h, i}
COMPLEMENT OF A SET:
          Find the following:
       A' is the shaded part below:
a. A - B = { a, b}
b. C - D = {  }
c. B - C = { c, d, e }
d. CARTESIAN PRODUCT -
The Cartesian product of set A and B,
written as A x B is the set of all possible
ordered pairs with first element from A and
second element from B:

             A x B = {( a, b ) / a ∈  A and b  ∈  B }

Example: Let A = { 2, 3, 5 } and B = { 7 , 8 } Examples:
Find each set ;
1. Using Venn Diagram find A' given  U
a. A x B = { ( 2, 7), ( 2, 8), ( 3, 7 ), ( 3, 8 ), = { 1 , 2, 3, 4, 5 }
( 5, 7 ), ( 5, 8 ) }
b. B x A = { ( 7, 2 ), ( 7, 3 ), ( 7, 5 ), ( 8, 2 ) ,
( 8, 3 ), ( 8, 5 )}
c. A x A = { ( 2, 2), ( 2, 3 ), ( 2, 5), ( 3, 2),
( 3, 3), ( 3, 5), ( 5, 2 ), ( 5, 3 ), ( 5, 5) }
2. Using English sentence description find source:https://www.google.com/url
A' Given U = all CE students and A = CE
students who are scholars. APPLICATION OF VENN DIAGRAM
If 380 students are taking courses: 215 taking
   Answer: A' = CE students who are not a Biology, 173 taking Physics, 182 taking
scholar.  chemistry.  72 taking Biology and Physics, 90
INTERSECTION OF SET taking Biology and Chemistry, 60 taking
Physics and Chemistry
A ∩ B is the shaded part below; assume the
sets are A and B: Find the number of students in each of the
following parts.

source:https://www.google.com/url
UNION OF SET
A∪B  is the shaded part below; assume the
Solution:
sets are A and B:
Let A = Biology ( 215)
       B = Physics ( 173)
       C = Chemistry ( 182 )
      A⋂B = 72
      A⋂C = 90
  The intersection of the three courses will be
source: https://www.google.com/url labeled as x.

DIFFERENCE OF SET Then...

A - B is the shaded part below A⋂B = 72 - x

A⋂C = 90 - x
B⋂C = 60 - x
Let us solve for the equation of A:
A = 215 - [ ( 72-x) +x +  (90-x)]
A = 215 - 162 + x
A= 53 + x
Let us proceed with B: A= { a, b, c, d } be the set of car brands
B =173 - [ ( 72 - x) + x + ( 60 - x )] B = { s, t, u, v } be the set of countries of the
car manufacturer.
B = 173 - (132 - x)
hence, A x B will give all the possible pairings
B = 41 + x of the elements of A and B.
Now let us have the C: then, the relation ( R) from A to B will be given
C =182 - [ ( 90 - x) + x + ( 60 - x )] by:

C = 182 - (90 - x + 60) R = { ( a, s ), ( a , t ), ( a , u ), ( a , v ), ( b , s ),

( b , t ), ( b, u ), ( b, v ), ( c , s ) , ( c, t ) , ( c , u ),
C = 32 + x ( b , v ) , ( d , s ), ( d , t ) , ( d , u ) , ( d , v ) }
Now let us find for x...   Let R be a relation from set A to the set B.
 53 + x + 72 - x + 41 - x + 90 - x + 60 - x + 32 -
 the domain of R is the set dom R.
x+x
348 + x = 380
         dom R = { a  ∈ A  ∣ ( a, b ) 
x = 32
∈ R for some b  ∈ B }
The value of x will be substituted to find the
answer to the Venn diagram...
 image ( or range ) of R.
A= 53 + 32 = 85
B = 41 + 32 = 73          im R = { 
R = { b ∈ B ∣ ( a , b ) ∈ R f o r s o m e a ∈ A }
C = 32 + 32 = 64
Example:
A⋂B = 72 - 32 = 40
    If  A = { 4, 7 }, then the relation from A to A
A⋂C = 90 - 32 = 58
will be A x A
B⋂C = 60 - 32 = 6           A x A = { ( 4, 4), ( 4 , 7), ( 7 , 4 ), ( 7 ,
1.4.3 LANGUAGE OF RELATIONS 7)}

Definition: 1.4.4 LANGUAGE OF FUNCTION

Definition:
 A relation is a set of ordered pairs.
 If x and y are elements of these sets  A function is a special kind of
and if a relation exists between x and y, relationship that helps visualize
then we say that x corresponds to y or that relationships in terms of graphs and make it
y depends on x and is represented as the easier to interpret different behavior of
ordered pair of point ( x, y ). variables.
 A relation from set A to set B is defined  A function is a relation in which, for each
to be any subset of A x B. value of the first component of the ordered
 If R is a relation from A to B and ( a, pairs, there is exactly one value of the
second component.
b )  ∈R, then we say that "a is related to  The set X is called the domain of the
b" and it is denoted as an R b. function.
 Let us take a look with this example:
 For each element of x in X, the There are several ways in finding an unknown
corresponding element y in Y is called the number.
value of the function at x, or the image of x.

Range - is the set of all images of the elements

of the domain. Number Pattern
  A function can be map one to one   This leads directly to the concept of
correspondence from one set to another. functions in mathematics about different
 A function can be map many to one quantities which are defined as the list of
correspondence from one set to another. the same numbers following a particular
sequence.
   It can also be applied to problem-solving
whether a pattern is present and can be used to
Domain ( x )                    Range ( f(x))
generalize a solution to a problem.

         a             ⟶                        s  The following are examples of number pattern;

         b               ⟶                     t   1.) Fibonacci sequence

  2.) Prime number sequence
         c           ⟶                         u
3.) Imaginary number sequence

         d         ⟶                         v 4.) geometric number pattern

  5.) growing number pattern.

 Examples:  In a given sequence we need to identify

whether it is infinite or finite...
    Determine whether each of the following
relations is a function or not a function: Hence, the difference between the two needs
to be considered.
1. A = { ( 1 , 3 ), ( 2 , 4 ), ( 3 , 5 ), ( 4 , 6 )} 
 Definition:
--- Function  
2. B = { ( - 2, 7), ( -1 , 3 ), ( 0 , 1 ), ( 1 , 5 ),
( 2 , 5 )} --- Function        An infinite sequence is a function
3. C = { ( 3 , 0), ( 3 , 2 ), ( 7, 4 ), ( 9 , 1 ) } whose domain is the set of positive
--- Function integers.

MODULE 2: MATHEMATICAL PROBLEM       The function values a1,  a2, a3, a4,...an-1, an...
SOLVING AND REASONING are the terms of the sequence.

2.1 NUMBER PATTERN         A finite sequence is a function

whose domain consists of the first n
Introduction positive integers only.  Take a look with
these examples:
Numbers are found everywhere in our daily
lives. Mathematics is based on numbers. Let us identify whether the following is a finite
or infinite sequence.
Mathematics is useful to predict and therefore
number pattern is about prediction.       1.) 1 , 2, 3, 4, 5, 6, 7 - Finite sequence
There are mathematical problems that involve        2.) 2, 4, 6, 8, 10, 12, 14 - Finite sequence
the number pattern.
       3.) 1, 3, 5, 7, 9, 11, 13, 15 -Finite                   S2 =  22 = 4
sequence
                  S3 = 32 = 9     therefore the first three
        4.) 3, 6, 9, 12,15...  - Infinite Sequence terms of the sequence are 1, 4, 9.
         5.) 1, 1, 2, 3, 5, 8, 13...- Infinite
Sequence 4. ) Pn =  1 21 2(3n2 - n )
         6.) 1, 4, 9,16,25, 36... - Infinite solution:  let n1=1, n2=2, n3=3
Sequence
Aside from the sequence stated above we also       then: p1 =   1 21 2(3(12) - 1 )=1
have the so-called General Sequence which
is in the form of a1, a2, a3, a4,...an-1, an...
                   P2 = 1 21 2 (3(22) - 2) = 5
This sequence has a1 as its first term, a2 as the
second term, a3 as the third term, and an as the
nth term which is also called the general                   P3 =  1 21 2(3(32) - 3) = 12 
term of the sequence. therefore the first three terms of the sequence
are 1, 5, 12.
 Let us take the following examples. 
5.) Hn = (2n2 - n )
Examples: Find the first three terms of the
sequence whose nth term is given by the solution:  let n1=1, n2=2, n3=3
formula:       then: H1 =  (2(12) - 1 )=1
1.) an = 3n+1                   H2 = (2(22) - 2) = 6
      solution:  let n1=1, n2=2, n3=3                   H3 = (2(32) - 3) = 15   therefore the
      then: a1 =  3(1) + 1 = 4 first three terms of the sequence are 1, 6, 15.

                  a2 = 3(2) + 1 = 7 2.2 DIFFERENCE TABLE

                  a3 = 3(3) + 1 = 10     therefore the Introduction:

first three terms of the sequence are 4, 7, 10.
The process of finding the next term in a given
sequence can also be found using another
2.) Tn =  1 21 2(n2 + n ) technique.
solution:  let n1=1, n2=2, n3=3 Finding the difference between the number
pattern will help us find the next term.
      then: T1 =   1 21 2(12 + 1 )=1 The difference table is needed to be
constructed.
                  T2 = 1 21 2 (22 + 2) =  3 DIFFERENCE TABLE
A difference table shows the difference
                  T3 =  1 21 2(32 + 3) = 6    between successive terms of the sequence.
therefore the first three terms of the sequence The differences in rows may be the first,
are 1, 3, 6. second, and third differences.  The following
examples will show how to predict the next
3.) Sn = n2
term of a sequence and we look for a pattern in
solution:  let n1=1, n2=2, n3=3 a row difference.

      then: S1 =  12 = 1 Examples: Construct the difference table to

predict the next term of each sequence.
1.) 3,7, 11, 15, 19, _____ The  first difference are not the same so lets
proceed in getting the second difference
solution:
5-3=2
 7 - 3 = 4
12 - 5 = 7
11 - 7 = 4
24 - 12 = 12
15 - 11 = 4
41 - 24 = 17
19 - 15 = 4
The second difference are not equal.
 Since the difference of the sequence are the
same this means that the first  difference in Proceed with the third difference
the  sequence is 4.
7-2=5
To find the next term we need to add 19 and 4
that are equal to 23.  12 - 7 = 5

hence, the next term in the sequence is 23... 17 - 12 = 5

2.) 4, 9, 17, 28, ______ The third difference is 5 add this to 17 so that
is 22, add 22 to the last difference in the first
solution: difference table which is 41 so the sum is 63. 
 9-4=5 63 will be added to the last term in the
sequence which is 91 to find the unknown
 17 - 9 = 8 term.
 28 - 17 = 11 Hence, the answer is 154.
The  first difference are not the same so let us 2.3 NUMBER SERIES
find the second difference:
Number Series is a sequence formed out of
8-5=3 numbers. The four fundamental operations
11 - 8 = 3 such as addition, subtraction, multiplication,
and division can be used to find the next term
The second difference are now the same, we in the given number series. The difference
are going to use 3 as our addend. table can also be used to find the next term in
the series.
The difference of 3 will be added to 11, and the
sum will be added to 28 to find the next term. Examples:
Hence, the next term will be 42. 1.) 2, 5, 10, 17, 26, 37, 50, ____

3. ) 6, 9 , 14, 26, 50, 91, ____   Solution:

5-2=3
solution:
10 - 5 = 5
 9 - 6 = 3
17 - 10 = 7
14 - 9 = 5
26 - 17 = 9
26 - 14 = 12
37 - 26 = 11
50 - 26 = 24
50 - 37 = 13
91 - 50 = 41
Let us find the second difference to find a
number pattern... 40  ÷÷÷ 4  = 10
5-3=2
10  ÷÷÷5 = 2
7-5=2
therefore the unknown term is 240.
9-7=2
5. ) 9, 17, 31, 57, _____, 205
11 - 9 = 2
13 - 11 = 2   9(2) - 1 = 17
Add 2 to 13 hence it is equal to 15... 17 ( 2) - 3 = 31
15 will be added to 50 and so the next term is 31 ( 2) - 5 = 57
65.
57 ( 2) - 7 = 107
2.) 2, 8, 16, 128, ______
107(2) - 9 = 205.
solution:
It important that in a number series one
2(8) = 16 should know how to detect the rules that will
result in the formation of a number.
8(16) = 128
2.4 WRITING A FORMULA FROM A
16(128) = 2048. SEQUENCE
Therefore the next term is 2048. Writing A Formula From A Sequence:
3.)120, 99, 80, 63, 48, _____ Examples;
solution; Determine the 100th term of the following.
12 x 10 = 120 1.)  7 , 10, 13, 16, _____, _____, ..._____
11 x 9   = 99       Solution:
10 x 8 = 80           Using the difference table does not apply
9 x 7 = 63 to this problem since the 100th term should be
found.
8 x 6 = 48
          We need to write a formula that will help
7 x 5 = 35 us solve this problem.
so the next term is 35.        Step 1: Find the difference in the pattern:
4.) 240, ____, 120, 40, 10, 2                         10 - 7 = 3
solution:                         13 - 10 = 3
                        16 - 13 = 3
240  ÷÷÷1 = 240
        Since 3 is the first difference we can have
an equation: 3n
240  ÷÷÷2 = 120
        Step 2:  Use the equation formulated in
step 1.
120  ÷÷÷3 = 40 
                  In 3n let n be any real number from       Step 1;   11 - (-1) = 12
1 to infinity.
                          31 - 11  = 20
                if n =1 then 3n = 3
                          59 - 31 = 28
                if n = 2 then 3n = 6
                          95 - 59 = 36
               if n = 3 then 3n = 9
          the first difference are not the same.
               if n = 4 then 3n = 12
          find the second difference.
        Step 3: Use the given sequence then
subtract the sequence formulated from 3n.                        20 - 12 = 8

                      7        10      13       16                        28 - 20 = 8

               -     3           6        9        12                        36 - 28 = 8

                      4            4       4         4   the         therefore the second difference is 8,

difference is +4 hence we will have a term of 8n2 this can be
reduced to 4n2.
        Step 4: Then the formula is 3n + 4.
  step 2: use 4n2 
        Step 5: Use the formula to solve for the
missing term.                  if n =1, then 4n2 = 4

                       3 ( 100) + 4 = 304 is the 100th                      n = 2, then 4n2 = 16

term                      n = 3, then 4n2 = 36
2. )  3 , 10 , 17, _____, ______...______.                      n = 4, then 4n2 = 64

 Solution:      then    - 1     11     31     59    95

     step 1:      10 - 3 = 7                 -     4     16     36    64    100

           17 - 10 = 7                       -5    -5     -5      -5     -5

      step 2:   use 7n the formula is 4n2 - 5 and the 100th term is
39,995.
        If n = 1 , then 7n= 7
2.5 SUMMARY OF TOPICS
            n = 2, then  7n = 14
Mathematics proved that it is useful in
            n = 3, then 7n = 21 predicting and number pattern is about
prediction.
             n = 4, then 7n = 28
Number pattern leads directly to the concept of
     Then  3    10    17       24        functions in mathematics about different
              -  7     14    21       28 quantities which are defined as a list of the
same numbers following a particular sequence.
              -4       - 4    - 4      -4
Number Pattern can also be applied to
 therefore the formula is  7n - 4 problem-solving whether a pattern is present
and can be used to generalize a solution to a
         The 100th term is 7(100) - 4 = 696. problem.
3. ) - 1, 11, 31, 59, 95, ....______. The difference table is a way of finding the next
number in a series.
 Solution:
The four basic operations can also be used to
find the missing term in a series of numbers. 
From a given series of numbers, one can
formulate a formula to help find succeeding
terms in a given sequence.
George Polya was a great mathematician that
introduces a unique way of solving problems.
The four steps in solving problem-solving by
George Polya are as follows;

1. Understand the Problem.

2. Make the Plan.
3. Do the Plan.
4. Look Back.