Mathematics 10

Quarter 1
Self-Learning Module 4
Sequence and Series
Mathematics – Grade 10
Quarter 1 – Self-Learning Module 4: Sequence and Series
First Edition, 2020

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Published by the Department of Education – Schools Division of Pasig City

Development Team of the Self-Learning Module

Writer: Leah F. Balajadia; Randell A. Diaz; Redentor J. Dizon III
Editor: Revie G. Santos
Reviewers: Revie G. Santos; Ma. Cynthia P. Badana; Ma. Victoria L. Peñalosa
Illustrator: Leah F. Balajadia
Layout Artist: Janeth D. Morte
Management Team: Ma. Evalou Concepcion A. Agustin
OIC-Schools Division Superintendent
Aurelio G. Alfonso EdD
OIC-Assistant Schools Division Superintendent
Victor M. Javeña EdD
Chief, School Governance and Operations Division and
OIC-Chief, Curriculum Implementation Division

Education Program Supervisors

Librada L. Agon EdD (EPP/TLE/TVL/TVE)

Liza A. Alvarez (Science/STEM/SSP)
Bernard R. Balitao (AP/HUMSS)
Joselito E. Calios (English/SPFL/GAS)
Norlyn D. Conde EdD (MAPEH/SPA/SPS/HOPE/A&D/Sports)
Wilma Q. Del Rosario (LRMS/ADM)
Ma. Teresita E. Herrera EdD (Filipino/GAS/Piling Larangan)
Perlita M. Ignacio PhD (EsP)
Dulce O. Santos PhD (Kindergarten/MTB-MLE)
Teresita P. Tagulao EdD (Mathematics/ABM)

Printed in the Philippines by Department of Education – Schools Division of

Pasig City
Mathematics 10
Quarter 1
Self-Learning Module 4
Sequence and Series
Introductory Message
For the Facilitator:

Welcome to the Mathematics Grade 10 Self–Learning Module on Sequence and

Series!

This Self-Learning Module was collaboratively designed, developed and

reviewed by educators both from the Schools Division Office of Pasig City headed by
its Officer-in-Charge Schools Division Superintendent, Ma. Evalou Concepcion A.
Agustin, in partnership with the City Government of Pasig through its mayor,
Honorable Victor Ma. Regis N. Sotto. The writers utilized the standards set by the K
to 12 Curriculum using the Most Essential Learning Competencies (MELC) in
developing this instructional resource.

This learning material hopes to engage the learners into guided and
independent learning activities at their own pace and time. Further, this also aims
to help learners acquire the needed 21st century skills especially the 5 Cs, namely:
Communication, Collaboration, Creativity, Critical Thinking and Character while
taking into consideration their needs and circumstances.

In addition to the material in the main text, you will also see this box in the
body of the self-learning module:

Notes to the Teacher

This contains helpful tips or strategies that
will help you in guiding the learners.

As a facilitator you are expected to orient the learners on how to use this self-
learning module. You also need to keep track of the learners' progress while allowing
them to manage their own learning. Moreover, you are expected to encourage and
assist the learners as they do the tasks included in the self-learning module.
For the Learner:

Welcome to the Mathematics Grade 10 Self–Leaning Module on Sequence and

Series!

This self-learning module was designed to provide you with fun and
meaningful opportunities for guided and independent learning at your own pace and
time. You will be enabled to process the contents of the learning resource while being
an active learner.

This self-learning module has the following parts and corresponding icons:

Expectations - This points to the set of knowledge and skills

that you will learn after completing the module.

Pretest - This measures your prior knowledge about the lesson

at hand.

Recap - This part of the module provides a review of concepts

and skills that you already know about a previous lesson.

Lesson - This section discusses the topic in the module.

Activities - This is a set of activities that you need to perform.

Wrap-Up - This section summarizes the concepts and

application of the lesson.

Valuing - This part integrates a desirable moral value in the

lesson.

Posttest - This measures how much you have learned from the
entire module.
 EXPECTATIONS

1. Define arithmetic sequence.

2. Identify an arithmetic sequence.
3. Give the next two terms of the given arithmetic sequence.

PRETEST

Directions: Choose among the choices which will be the next two terms of each
sequence below. Write the letter that corresponds to the correct answer.

1. 8, 15, 22, 29, 36, _____, _____

A. 42, 49 B. 43, 50 C. 44, 51 D. 45, 52

2. 21, 17, 13, 9, 5, _____, _____

A. 1, −3 B. −9, −13 C. 9, 13 D. 0, −4

3. −32, −29, −26, −23, −20, _____, _____

A. −18, −15 B. 18, 15 C. −17, −14 D. 17, 14

4. 5, 2, −1, −4, −7, _____, _____

A. −12, −15 B. −11, −14 C. −10, −13 D. −9, −12

5. −28, −22, −16, −10, −4, _____, _____

A. 1, 7 B. 0, 6 C. −2, −8 D. 2, 8

RECAP

Directions: Perform the indicated operation. Write your answer on the space
provided.

1. 3 + (−5) = _______
2. −3 + (−5) = _______
3. −3 − 5 = _______
4. 3 − (−5) = _______
5. −3 − (−5) = _______
 LESSON

Sequence

A sequence is a set of numbers that follows each other in a logical manner.

Types of Sequences

1. Arithmetic Sequence
2. Geometric Sequence
3. Harmonic Sequence
4. Fibonacci Sequence

Let’s have one example of a sequence.

The ground floor of the Pasig City Hall

is 6-meter high and each floor above it is 4-
meter high. If the Pasig City Hall has 8 floors,
how high is it?

Solution:

The measure of the first floor of the city

hall serves as the first term of the sequence.
Then, we add 4 to get the next term. We call 4
as the common difference of the sequence.
The common difference is the number we add
to the previous term so that we get the next
terms.

Continuing the sequence, you will get 𝑎8 = 34.

Therefore, the height of the 8-storey Pasig City Hall is 34 meters.

The problem given above is an example of an arithmetic sequence.

Arithmetic Sequence

Arithmetic sequence is a sequence of numbers such that the difference

between the consecutive terms is constant.
 We call the distance between the terms of an arithmetic sequence as common
difference (d).

Examples:

A. Identify if the set of numbers is an arithmetic sequence or not.

1) 2, 5, 8, 11, 14, …

We will know if the given sequence is arithmetic if there is a common

difference. So, for this example, we will subtract the first term 2 from the second term
5, the second term 5 from the third term 8, and so on, to see if there is a common
difference.

5−2=3 11 − 8 = 3

8−5=3 14 − 11 = 3

The common difference is 3. So, the given sequence is an arithmetic

sequence.

2) 1, 2, 4, 7, 11, …
Let us get the common difference.

2−1=1 7−4=3

4−2=2 11 − 7 = 4

Looking at the solution, we could see no common difference. Hence, the given
is not an arithmetic sequence.

3) 0, 4, 8, 12, 16, …
Again, we need to find the common difference.

4−0=4 12 − 8 = 4

8−4=4 16 − 12 = 4

We obtained 𝑑 = 4. So, the given sequence is an arithmetic sequence.

4) 41, 35, 29, 23, 17, …

Subtract the consecutive terms to get the common difference.

35 − 41 = −6 23 − 29 = −6

29 − 35 = −6 17 − 23 = −6

Here, 𝑑 = −6. Therefore, it is an arithmetic sequence.

5) 1, 1, 2, 3, 5, …

Let us find the common difference.

1−1=0 3−2=1

2−1=1 5−3=2
 Since there is no common difference in this sequence, therefore it is not an
arithmetic sequence.

B. Give the next two terms of each of the given sequences below.
1) 1, 5, 9, 13, 17, _____, _____.

To get the next term of an arithmetic sequence, we need to get first the
common difference. So, for this example,

5−1=4 13 − 9 = 4

9−5=4 17 − 13 = 4

The common difference, 𝑑 = 4.

Next, we need to add the common difference to the previous term to get the
next term.

17 + 4 = 21

21 + 4 = 25

So, the next two terms are 21 and 25.

2) 7, 9, 11, 13, 15, _____, _____.

Let us find the common difference.

9−7=2 13 − 11 = 2

11 − 9 = 2 15 − 13 = 2

We have 𝑑 = 2. Next, let’s find the next two terms by adding the common
difference to the previous term.

15 + 2 = 17

17 + 2 = 19

Therefore, the next two terms are 17 and 19.

3) −8, −5, −2, 1, 4, _____, _____.

Solve for the common difference.

−5 − (−8) = 3 1 − (−2) = 3

−2 − (−5) = 3 4−1=3

The common difference, 𝑑 = 3. Now, let us get the next two terms.

4+3=7

7 + 3 = 10

The next two terms are 7 and 10.

 4) 51, 43, 35, 27, 19, _____, _____.

Find the common difference.

43 − 51 = −8 27 − 35 = −8

35 − 43 = −8 19 − 27 = −8

The common difference is −8. We can now get the next two terms.

19 + (−8) = 11

11 + (−8) = 3

So, the next two terms are 11 and 3.

5) 5, 1, −3, −7, −11, _____, _____.

Solve for the common difference.

1 − 5 = −4 −7 − (−3) = −4

−3 − 1 = −4 −11 − (−7) = −4

The common difference, 𝑑 = −4. Now, let us get the next two terms.

−11 + (−4) = −15

−15 + (−4) = −19

So, the next two terms are −15 and −19.

ACTIVITIES

ACTIVITY 1: LET’S PRACTICE!

Direction: Write AS if the given sequence is an arithmetic sequence and NOT if it is
not arithmetic.
1. 1, 8, 27, 64, 125, …
2. 23, 27, 31, 35, 39, …
3. 18, 15, 12, 9, 6, …
4. −2, −1, 1, 4, 8, …
5. −7, −9, −11, −15, −17, …
ACTIVITY 2: KEEP PRACTICING!
Direction: Write the next two terms of each of the following arithmetic sequences
below.
1. 17, 20, 23, 26, 29, _____, _____.
2. −6, −2, 2, 6, 10, _____, _____.
3. 14, 11, 8, 5, 2, _____, _____.
4. 24, 31, 38, 45, 52, _____, _____.
5. 3, 1, −1, −3, −5, _____, _____.

ACTIVITY 3: TEST YOURSELF!

A. Direction: Write AS if the given sequence is an arithmetic sequence and NOT
if it is not arithmetic.
1. 17, 20, 23, 26, 29, …
2. 1, 1, 2, 3, 5, …
3. 18, 10, 2, −6, −4, …

B. Direction: Write the next two terms of each arithmetic sequence below.
1. 43, 38, 33, 28, 23, _____, _____.
2. 2, −2, −6, −10, −14, _____, _____.

WRAP–UP

How do we find the next term in an arithmetic sequence?

1. Identify first if the sequence is arithmetic by getting a common difference

between the consecutive terms.
2. To get the next term, simply add the common difference (d) to the previous
term.
 VALUING

Reflection: (Journal Writing)

As a student, share your thoughts and insights about the steps that you
want to take to achieve your dreams in life.

POSTTEST

Directions: Read each statement carefully. Write the letter that corresponds to the
correct answer.
1. Is the sequence 18, 24, 30, 36, 42, … an arithmetic sequence?

A. Yes B. No C. Maybe D. None of these

2. Is the sequence 15, 6, −3, −12, −21, … an arithmetic sequence?

A. Yes B. No C. Maybe D. None of these

3. Is the sequence 4, 1, −4, −11, −20, … an arithmetic sequence?

A. Yes B. No C. Maybe D. None of these

4. What will be the next term of the arithmetic sequence 22, 30, 38, 46, 54?

A. 60 B. 61 C. 62 D. None of these

5. What will be the next two terms of the arithmetic sequence 12, 2, −8, −18, −28?

A. 38, 48 B. −32, −22 C. 8, 18 D. None of these

 1. A
2. A
3. C
4. C
5. D
POSTTEST
A. 1. AS 1. 32, 35 1. NOT
2. NOT 2. 14, 18 2. AS
3. NOT 3. −1, −4 3. AS
B. 1. 18, 13 4. 59, 66 4. NOT
2. −18, −22 5. −7, −9 5. NOT
ACTIVITY 3 ACTIVITY 2 ACTIVITY 1
1. −2 1. B
2. −8 2. A
3. −8 3. C
4. 8 4. C
5. 2 RECAP 5. D PRETEST
KEY TO CORRECTION
 REFERENCES

Callanta, Melvin, Canonigo, Allan, and Arnaldo Chua. Mathematics Learner’s Module
10. Manila: Department of Education, 2015.

Nivera, Gladys, and Minie Rose Lapinid. Grade 10 Mathematics: Patterns and
Practicalities K – 12. Manila: Salesiana Books by Don Bosco Press, Inc., 2015.

Oronce, Orlando, and Marilyn Mendoza. E – Math 10: Worktext in Mathematics.

Manila: Rex Book Store, Inc., 2017.

Oronce, Orlando, and Marilyn Mendoza. E – Math II: Intermediate Algebra, Worktext
in Mathematics for Second Year High School. Manila: Rex Book Store, Inc.,
2007.

Oronce, Orlando, and Marilyn Mendoza. Exploring Mathematics II: Intermediate

Algebra. Manila: Rex Book Store, Inc., 2003.

Pasig City Hall image.

http://admin2018.pasigcity.gov.ph//uploads/cityhall-
images/IMG201809051022202018918_192243.jpg. (accessed July 1, 2020.)