Introduction to Sequences and Series.pptx

Patterns&Sequences:
AnIntroduction to Sequences&
Series
Dr. Michelle P. Bales
Math 10_Lesson 1.1_mcpbales
1

What is common about the pictures?
2

Patterns & Sequence in Nature
Sequences can be observed from how the petals and leaves of plants are
arranged. The growth of plants and human follow a certain stage and
pattern.
3

Sequence in daily life
Sequences can also be observed on how we arrange our things and on the
ways we do our stuff like in sports or our daily routine.
4

Sequences in real life
Sequences can likewise be seen on the patterns of how infection spreads,
on the arrangement of cables in a computer, up to the sequence of the DNA,
which makes us what we are.
5

Let us learn
patterns and
sequences.
6

M10A
Learning Competency
The learner demonstrates understanding of key concepts of
sequences, polynomials and polynomial equations.
Content Standard
The learner is able to formulate and solve problems
involving sequences, polynomials and polynomial equations
in different disciplines through appropriate and accurate
representations.
Learning Competency
The learner generates patterns. (M10AL-Ia-1 )
Specific Objectives
At the end of the week, students must have
1) classified sequences as finite or infinite
2) identified the terms in a sequence;
3) generated patterns.

FLOW OF THE SESSION
1
CONCEPT
2
EXERCISES
3
ONLINE LINKS
8
3
QUIZ
Study Sequences and
Series
Answer the exercises
in your notebook and
check your own work.
Answer Quizizz and/or
Khan Academy
Exercises.
Answer the Quiz in
Edmodo.

9
SEQUENCE
-
a set of things (usually numbers)
that are in order.
Example
2, 4, 6, 8, 10, 12, … , 2n
1ST term
a1
2ND term
a𝟐
3RD term
a3
The three dots is an ellipsis which indicates that
the list goes on.
last term/ nth term/general term
Note: There must be a
pattern in which these
numbers or objects
are organized.

10
that are in order.
How are these
sequences generated?
2) 3, 10, 17, 24, 31, …
3) 2, 4, 8, 16, 32, …
4)
3
2
,
5
4
,
7
6
,
9
8
, …

11
that are in order.
2) 3, 10, 17, 24, 31, …
3) 2, 4, 8, 16, 32, …
4)
3
2
,
5
4
,
7
6
,
9
8
, …
seven is added to get the next term
multiply by 2 to get the next term
add two to both the numerator and the denominator to get the next term
How are these
sequences generated?

12
that are in order.
Can you give the next
three terms of the
sequence?
2) 3, 10, 17, 24, 31, …
3) 2, 4, 8, 16, 32, …
4)
3
2
,
5
4
,
7
6
,
9
8
, …

13
that are in order.
Can you give the next
three terms of the
sequence?
2) 3, 10, 17, 24, 31,
3) 2, 4, 8, 16, 32,
4)
3
2
,
5
4
,
7
6
,
9
8
,
38, 45, 52
64, 128, 256
𝟏𝟏
𝟏𝟎
,
𝟏𝟑
𝟏𝟐
,
𝟏𝟓
𝟏𝟒

14
that are in order.
How are these
sequence generated?
5) A, D, G, J, M, P, …
6) , , , , , …
7) , , , ,

15
that are in order.
How are these
sequence generated?
5) A, D, G, J, M, P, …
6) , , , , , …
7) , , , ,
Skip two letters in the alphabet to get the next term
Follows the pattern circle, square, triangle,
The circle follows the pattern “hollow, filled” and the arrows are rotated counter clockwise 450 at a time.

16
that are in order.
Can you give the
next three terms in
the sequence? Write
your answer in your
notebook.
5) A, D, G, J, M, P,
6) , , , ,
7) , , ,

17
that are in order.
Can you give the
next three terms in
the sequence? Write
your answer in your
notebook.
5) A, D, G, J, M, P,
6) , , , , , , ,
7) , , ,
S, V, Y

18
Exercise #1. Give the next three terms of the sequence.
Write your answer in your notebook.
1) 2, 7, 12, 17, 22, ___, ___, ___
2) 0.5, 1, 1.5, 2, 2.5, ___, ___, ___
3) 7, 8, 15, 23, 38, ___, ___, ___
4) Z1, Y2, X3, W4, V5, ___, ___, ___
5) , , , , ___, ___, ___

19
Exercise #2. Write your answer in your notebook.
Formulate a sequence with 5 terms where the next term is
one more than twice the current term. You may select any
number as the first term.

20
Example:
Sequence First Term Last Term
Number of
Terms
2, 4, 6, 8, 10 2 10 5
-5, -3, -1, …., 17, 19 -5 19 13
60, 50, 40, 30, 20, 10 60 10 6
The first and the last terms are referred to as
extremes and those in between them are
called means.

21
Example:
3, 6, 9, 12, 15, …
64, 32, 16, 8, 4, …
…, -2, -1, 0, 1, 2, …
The three dots (ellipsis)
indicates that the pattern
continues on that side. Since no
number is written after it, it
means it goes on without end.

Can you give an example, with 5 terms, of
22
• a finite sequence
• an infinite sequence

Which of the following sequences are finite?
1, 4, 7, 10, ..., 37, 40
23
... , -14, -7, 0, 7, 14, ...
Apr, June, Sept, Nov
1, 10, 101, 1010, ...
A
C
B
D

24
SEQUENCE – is a function whose domain is the set of
natural numbers or a subset of consecutive positive integers.
Natural Number are
the set of counting
numbers.
Example 8:
Use the functional Notation F(n) = 2n – 3, where n is a natural number, to write an
infinite sequence.
If n=1
F(1) = 2(1) – 3
= 2 – 3
= -1
If n=2
F(2) = 2(2) – 3
= 4 – 3
= 1
If n=3
F(3) = 2(3) – 3
= 6 – 3
= 3
If n=4
F(4) = 2(4) – 3
= 8 – 3
= 5
The sequence is {-1, 1, 3, 5, … }.

25
SEQUENCE – is a function whose domain is the set of
natural numbers or a subset of consecutive positive integers.
Example 9:
Using the consecutive positive integers n = 1, 2, 3, 4, 5, write the first five terms of the
sequence defined by G n = n −
1
𝑛
.
If n=1
G 1 = 1 −
1
1
= 1 – 1
= 0
If n=2
G 2 = 2 −
1
2
=
4
2
-
1
2
=
3
2
If n=3
G 3 = 3 −
1
3
=
9
3
-
1
3
=
8
3
If n=4
G 4 = 4 −
1
4
=
16
4
-
1
4
=
15
4
The first five terms of the sequence are {0,
3
2
,
8
3
,
15
4
,
24
5
}
If n=5
G 5 = 5 −
1
5
=
25
5
-
1
5
=
24
5

26
Exercise #3. Find the first five terms of the sequence
given the nth term. Write your answer in you notebook.
1) an= n + 4
2) an= 2n – 1
3) an= 12 – 3n
4) an= 3n
5) an= (-2)n

27
Be sure you answered the exercises honestly before
checking. Please be honest, if you got wrong write the
corrections to remind you of your mistake and NOT to
commit the same mistake next time.
Write your score on top of every exercise.

28
Exercise #1. Give the next three terms of the sequence.
Write your answer in you Lecture Notebook.
1) 2, 7, 12, 17, 22, 27, 32, 37
2) 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4
3) 7, 8, 15, 23, 38, 61, 99, 160
4) Z1, Y2, X3, W4, V5, U6, T7, S8
5) , , , , , ,
One point per term.
15 points in all.

29
Exercise #2. Write your answer in your Notebook.
Formulate a sequence with 5 terms where the next term is
one more than twice the previous term. You may select any
number as the first term.
(Answers may vary depending on the first term you chose.)
Sample answer: 1, 3, 7, 15, 31 or 2, 5, 11, 23, 47
One point per term.
5 points in all.

30
Exercise #3. Find the first five terms of the sequence
given the nth term. Write your answer in your notebook.
1) an= n + 4
2) an= 2n – 1
3) an= 12 – 3n
4) an= 3n
5) an= (-2)n
1) 5, 6, 7, 8, 9
2) 1, 3, 5, 7, 9
3) 9, 6, 3, 0, -3
4) 3, 9, 27, 81, 243
5) -2, 4, -8, 16, -32
One point per term.
25 points in all.

31
Please take quiz #1 in
our
Edmodo classroom.
GOOD LUCK!!!

Introduction to Sequences and Series.pptx

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