quarter 1 module 2: arithmetic sequence
TRANSCRIPT
Mathematics
Quarter 1 – Module 2:
Arithmetic Sequence (M10ALIb-1)
10
Mathematics – Grade 10 Self-Learning Module (SLM) Quarter 1 – Module 2: Arithmetic Sequence First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.
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Office Address: Regional Center, Brgy. Carpenter Hill, City of Koronadal
Telefax: (083) 2288825/ (083) 2281893
E-mail Address: [email protected]
Development Team of the Module
Writer: Bainalyn G. Abas
Editor (Language/Social Content): Vivencio O. Aniñon, Ed.D/Ruby A. Buhat, Ed.D
Reviewer: Nora B. Mendoza/ Maureen Socorro N. Muñasque
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Concepcion F. Balawag, CESO V - Schools Division Superintendent
Edgar S. Sumapal - Assistant Schools Division Superintendent
Gilbert B. Barrera – Chief, CLMD
Arturo D. Tingson Jr. – REPS, LRMS
Peter Van C. Ang-ug – REPS, ADM
Jade T. Palomar - REPS, Mathematics
Pancho G. Balawag, Ed. D - CID Chief
Engr. Reynaldo SE Villan - EPS In Charge of LRMS
Vivencio O. Aniñon, Ed.D - Division ADM Coordinator
Engr. Reynaldo SE Villan – EPS, Math
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Mathematics Quarter 1 – Module 2:
Arithmetic Sequence
(M10ALIb-1)
2
Introductory Message
For the facilitator:
Welcome to the (Mathematics 10 ) Self-Learning Module (SLM) on Arithmetic
Sequence !
This module was collaboratively designed, developed and reviewed by educators both
from public and private institutions to assist you, the teacher or facilitator in helping
the learners meet the standards set by the K to 12 Curriculum while overcoming
their personal, social, and economic constraints in schooling.
This learning resource hopes to engage the learners into guided and independent
learning activities at their own pace and time. Furthermore, this also aims to help
learners acquire the needed 21st century skills 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 module:
As a facilitator you are expected to orient the learners on how to use this module.
You also need to keep track of the learners' progress while allowing them to manage
their own learning. Furthermore, you are expected to encourage and assist the
learners as they do the tasks included in the module.
Notes to the Teacher
This contains helpful tips or strategies that
will help you in guiding the learners.
3
For the learner:
Welcome to the Mathematics 10 Self-Learning Module (SLM) on Arithmetic Sequence
!
The hand is one of the most symbolized part of the human body. It is often used to
depict skill, action and purpose. Through our hands we may learn, create and
accomplish. Hence, the hand in this learning resource signifies that you as a learner
is capable and empowered to successfully achieve the relevant competencies and
skills at your own pace and time. Your academic success lies in your own hands!
This 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 module has the following parts and corresponding icons:
What I Need to Know
This will give you an idea of the skills or
competencies you are expected to learn in the
module.
What I Know
This part includes an activity that aims to
check what you already know about the
lesson to take. If you get all the answers
correct (100%), you may decide to skip this
module.
What’s In
This is a brief drill or review to help you link
the current lesson with the previous one.
What’s New
In this portion, the new lesson will be
introduced to you in various ways such as a
story, a song, a poem, a problem opener, an
activity or a situation.
What is It
This section provides a brief discussion of the
lesson. This aims to help you discover and
understand new concepts and skills.
What’s More
This comprises activities for independent
practice to solidify your understanding and
skills of the topic. You may check the
answers to the exercises using the Answer
Key at the end of the module.
What I Have Learned
This includes questions or blank
sentence/paragraph to be filled in to process
what you learned from the lesson.
What I Can Do
This section provides an activity which will
help you transfer your new knowledge or skill
into real life situations or concerns.
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Assessment
This is a task which aims to evaluate your
level of mastery in achieving the learning
competency.
Additional Activities
In this portion, another activity will be given
to you to enrich your knowledge or skill of the
lesson learned. This also tends retention of
learned concepts.
Answer Key
This contains answers to all activities in the
module.
At the end of this module you will also find:
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities
included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are
not alone.
We hope that through this material, you will experience meaningful learning and
gain deep understanding of the relevant competencies. You can do it!
References This is a list of all sources used in developing
this module.
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What I Need to Know
This module was designed and written with you in mind. It is here to help you master
the arithmetic sequence and other concepts related to it. The scope of this module
permits it to be used in many different learning situations. The language used
recognizes the diverse vocabulary level of students. The lessons are arranged to follow
the standard sequence of the course. But the order in which you read them can be
changed to correspond with the textbook you are now using.
The module is divided into three lessons, namely:
Lesson 1 – Definition of Arithmetic Sequence
Lesson 2 – Common Difference
Lesson 3 – General Term of an Arithmetic Sequence
After going through this module, you are expected to:
1. define arithmetic sequence;
2. identify the succeeding term in the sequence;
3. determine the common difference of an arithmetic sequence;
4. write the first five terms of a sequence;
5. generate a general term of the given arithmetic sequence.
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Let us check what you already know about arithmetic sequence by answering the
question below.
1. What is the next term in the arithmetic sequence 7, 15, 23, 31, ____?
A. 35 B. 32 C. 39 D. 41
2. Which of the following is an example of an arithmetic sequence?
A. ½, ¼, 1/6, 1/8,… B. 3, 5, 7, 9, 11,…
C. 2, 6, 18, 54,… D. 64, 32, 16, 8,…
3. The general term for the arithmetic sequence 9, 15, 21, 27, 33,... is given as
__________.
A. an = 9 + (n-1)d B. an = n + 9d
C. an = 9 + nd D. an = 9d + n
4. Find the first five terms of the sequence an = 4 + 5n.
A. 4, 8, 12, 16, 20 B. 25, 20, 15, 10, 5
C. 9, 14, 19, 24, 29 C. 4, 6, 8, 12, 14
5. Ellen earns P200 on the first day of work and additional P50 on every succeeding
day. How much is his salary in the fourth day?
a. P300 b. P350 c. P400 d. 450
6. Which of the following sequences follow the general term an = 2n - 1?
A. 9, 11, 13, 15,… B. 3, 6, 9, 12, 15,…
C. 2, 4, 6, 8, 10,… D. 4, 8, 12, 16, 20,…
7. The term between 8 and 24 is _______.
A. 12 B . 16 C. 18 D. 24
8. What is the common difference in the arithmetic sequence 7, 15, 23, 31,…?
A. 10 B. -10 C. 8 D. -8
9. What number is added every after each number to obtain the succeeding
numbers in the sequence 72, 60, 48, 36,….?
A. 12 B. -12 C. 24 D. -24
10. The next term for the sequence ½, 1, 3/2, 2 is ______.
A. ¾ B. 5/3 C. 5/2 D. 4
What I Know
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For numbers 11-15, tell whether the statement is TRUE or FALSE.
___________11. A sequence can be finite or infinite.
___________12. Arithmetic sequence is type of sequence where every term after the
first is obtained by adding a constant called the common difference.
___________13. The next term for the sequence 12, 18, 24 is 36.
___________14. The common difference in the arithmetic sequence 95, 83, 71, 59,
47, 35 is 12.
___________15. 1, 4, 7, 10, 13, … is an example of arithmetic sequence.
Very well done, be prepared for an exciting journey that awaits!
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Lesson
2 Arithmetic Sequence
Another module is set for you to explore. Patterns are everywhere and some
follow a specific order. Are you excited to know more about this? After going through
this module, you are expected to illustrate the arithmetic sequence as the first type
of sequence.
What’s In
Previously, you have learned about patterns that are present in sets and in
numbers. Now. Let us try to look back and relate it to our discussion.
Consider the following figure and sets of number below.
a.
b. 0, 5, 10, 15, 20, 25, 30,….
c. 3, 6, 12, 18, 36, 72,…
d. 160, 80, 40, 20, 10,….
Identifying and extending patterns are key concept in understanding
arithmetic sequence. Previously, we have learned that sequence is a pattern of
numbers where order matters. It can be finite or infinite.
Were you able to identify the succeeding term or figure? Is there a pattern in the
following sets?
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A sequence is a function whose domain is the finite set {1, 2, 3,…, n} or the
infinite set {1, 2, 3,… }.
Example:
n 1 2 3 4 5
an 3 -1 1.5 10 𝜋
The finite sequence has 5 terms. We may use the notation a1, a2, a3,…,an to denote
a(1), a(2), a(3),…,a(n) respectively.
In Grade 10, we often encounter sequences that form a pattern such as that found
in the sequence below.
Example:
n 1 2 3 4 5
an 4 7 10 13 16
The above sequence is an infinite sequence where an=3n+1.
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What’s New
This module stresses on the understanding of the nature of sequence and illustrating
arithmetic sequence. Study the problem below and answer the questions that follow.
Questions:
1. How much did he earn in November? ________________________
2. How much did he gain after five months? ____________________
3. How much is his gain in August? ______________________________
4. How much is the additional profit per month?__________________
5. Is there a pattern on the profits that Mr. Ahmer gains? _________
Very Good! Now, you are heading for more learnings on arithmetic sequence.
What is It
Below are important terminologies, notations and symbols that you must learn and
remember about arithmetic sequence.
An arithmetic sequence is a sequence where every term after the first is
obtained by adding a constant called the common difference.
Examples:
a. Is the pattern 9, 16, 23, 30, 37,… an arithmetic sequence?
The answer is yes because every term after 9 is obtained by adding 7.
a1=9 a2=9+7=16 a3=16+7=23 a4=23+7=30
a5=30+7=37
b. What is the next term for the arithmetic sequence 99, 90, 81, 72,
63,____?
Mr. Ahmer owns a business establishment and gains a profit every after another
month. He gained P7,250 in September, P8,360 in October and increases
similarly for the next months.
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Examples:
1. Give the first five terms of the
arithmetic sequence an=-3n+5.
a1 = -3(1) + 5 = -3 + 5 = 2
a2 = -3(2) + 5 = -6 + 5 = -1
a3 = -3(3) + 5 = -9 + 5 = -4
a4 = -3(4) + 5 = -12 + 5 = -7
a5 = -3(5) + 5 = -15 + 5 = -10
The first five terms of the
arithmetic sequence an = -3n + 5
are 2, -1, -4, -7 and -10.
2. Write the general term for the sequence
13, 25, 37,49, 61, …
Let us now form two equations from the
following succeeding term using the formula
ax + b = 0 where x stands for n.
a(n) + b = an 3a + b = 37
- ( 2a + b = 25)
a = 12
3(12) + b = 37
36 + b = 37
b = 37- 36
b=1
Substituting the values of a and b
12n + 1 = an or an = 12n + 1
The common difference is the constant amount of change between numbers in an
arithmetic sequence.
Examples:
a. What is the common difference in the 18, 25, 32, 39, 46?
The answer is 7 since it is the number constantly added to each term to
obtain the succeeding number.
46-39=7 39-32=7 32-25=7 25-18=7
b. Find the common difference in the arithmetic sequence 121, 99, 88,
77, …
The answer is -11 since each term in the sequence obtained by adding
this number every after the other.
77-88=-11 88-99=-11 99-121=-11
c. Identify the common difference for the sequence 9, 18, 27, 36, 45,…
The answer is 9 since it is the common number being added to the first
term and the succeeding terms.
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What’s More
Activity 1.1 Understanding Science Words
Let us try to answer more challenging set of problems and activities about
arithmetic sequence.
A. We need matchsticks for this activity.
1. Below are squares formed by matchsticks.
2. Count the number of matchsticks in each figure and record the results in a
table.
number of squares 1 2 3 4 5 6 7 8 9 10
number of match sticks
Questions:
1. What is the difference between the number of matchsticks used of one square
and two squares? _____________________
2. How many sticks will be used to form five squares? ______________________
3. Write the general term for the given sequence formed by the number of
matchsticks. _________________________________________
B. Supply the first five terms given the following general term.
1. an = 5n - 7 _________________________________________________
2. an = -3n + 13 _________________________________________________
C. Write the general term for the sequence -3, 8, 19, 30, 41,…
_____________________________________________________
You are great! Be prepared for the next battles.
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What I Have Learned
Here is another activity that lets you apply what you learned about illustrating
arithmetic sequence. Fill in the blanks with the correct word/s or symbol that best
fits the statement.
What I Can Do
Here is another activity that lets you apply what you learned about illustrating
arithmetic sequence Fill in the blanks with the correct word/s or symbol that best
fits the statement.
In mathematics, ___________ is an enumerated collection of objects in which
repetitions are allowed and order does matter. It can be _______ or ___________.
An ____________________ is a sequence where every term after the first is
obtained by adding a constant called the common difference. Thus
____________________ is then constant amount of change between numbers in an
arithmetic sequence.
For instance in the sequence -12, -5, 2, 9, 1,… the common difference is
___________ and the general term is __________________________________.
In mathematics, ___________ is an enumerated collection of objects in which
repetitions are allowed and order does matter. It can be _______ or ___________.
An ____________________ is a sequence where every term after the first is
obtained by adding a constant called the common difference. Thus
____________________ is then constant amount of change between numbers in an
arithmetic sequence.
For instance in the sequence -12, -5, 2, 9, 1,… the common difference is
___________ and the general term is __________________________________.
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Assessment
I hope you find it great going over this module. For you to determine how much
you’ve learned, please answer the questions by choosing the letter of the best
answer.
1. What is the next term in the arithmetic sequence -14, -8, -2, 4, ____?
A. -10 B. 12 C. -8 D. 10
2. Which of the following is an example of an arithmetic sequence?
A. 2/5, 2/7, 2/9, 2/11,… B. 3, 1, -1, -3, -5, -7,…
C. 1, 2, 4, 8, 16,… D. 54, 18, 6, 2,…
3. The general term for the arithmetic sequence 11,18, 25, 32,... is given as
__________.
A. an = 4n - 7 B. an = 4n + 7
C. an = 7n + 4 D. an = 7n – 4
4. Find the first five terms of the sequence an = 3n - 11
A. 8, 11, 14, 17,20 B. -8, -5, -2, 1, 4
C. 3, 6, 9, 12, 15 C. 11, 22, 33, 44, 55
5. Ryan plants 38 tree seedlings on Saturday. On the Sunday day he finished
planting 50 tree seedlings and add the same amount everyday other day. How
many seedlings did he planted on Wednesday?
A. 79 B. 96 C. 86 D. 87
6. Which of the following sequences follows the general term an=3n-5?
a. 3, 6, 9, 12, 15,… c. -2, 1, 4, 7, 11,…
b. 5, 10, 15, 20, 25,… d. 5, 8, 11, 14, 17,…
7. The term between 5 and 15 is _______.
A. 10 B. 11 C. 13 D. 9
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8. What is the common difference in the arithmetic sequence -32, -21, -10, 1, 12,?
A. 11 B. -11 C. 12 D. -12
9. What number is added every after each number to obtain the succeeding
numbers in the sequence 9, 15, 21, 27,….?
A. 8 B. 7 C. 6 D. -5
10. The next term for the sequence 1/3, 2/3, 1, 4/3, 5/3 is ______.
A. 1 B. 2 C. 3 D. 4
For numbers 11-15, tell whether the statement is TRUE or FALSE.
___________11. A sequence is always infinite.
___________12. Arithmetic sequence is type of sequence where every term after the
first is obtained by multiplying a constant called the common
difference.
___________13. The next term for the sequence -13, 26, 65, 104 is 143.
___________14. The common difference in the arithmetic sequence 3 789, 3 664,
3 539, 3 414,… is 125. .
___________15. 3, 6, 18, 54, ,… is an example of arithmetic sequence.
Congratulations for the job well done! Just keep the faith.
Additional Activities
Try this problem to boost your understanding on arithmetic sequence.
After a knee surgery, your trainer tells you to return to your jogging program
slowly. He suggests jogging for 12 minutes each day for the first week. Each week,
thereafter, he suggests that you increase that time by 6 minutes per day. On what
week will it be before you are up to jogging 60 minutes per day?
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Answer Key
What I Know
1. C
2. B
3. A
4. C
5. B
6. B
7. B
8. C
9. B
10. C
11. TRUE
12. TRUE
13. FALSE
14. FALSE
15. TRUE
Whats In
1.
2. 35
3. 144
4. 5
Whats New
1. He earned P9479 in
November.
2. He earned a total of P47 350 in five months.
3. He gained P6 1440 in August.
4. His additional profit
per month is P1 110.
5. Yes, each month the
gain continuously
increases by P1 110.
17
Whats More A.
number of
squares
1 2 3 4 5 6 7 8 9 10
number of match
sticks
4 7 10 13 16 19 22 25 28 31
1. The difference is 3. 2. We need 16 mathsticks.
3. an=4+3n
B.
1. -2, 3, 8, 13, 18
2. 16, 19, 22, 25, 28
C. an=11n-14
What I Have Learned
In mathematics, sequence is an enumerated collection of objects in which repetitions
are allowed and order does matter. It can be finite or infinite.
An arithmetic sequence is a sequence where every term after the first is
obtained by adding a constant called the common difference. Thus common
difference is then constant amount of change between numbers in an arithmetic
sequence.
For instance in the sequence -12, -5, 2, 9, 1,… the common difference is 7 and
the general term is an=7n-19.
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What Can I Do
a.
Day 1 2 3 4 5
No Of
Participants
20 30 40 50 60
b. an= 10n+10 a1=10(1)+10=20 a2=102)+10=30
c. 90
Assesment
1. D
2. B
3. C
4. C
5. B
6. C
7. A
8. A
9. C
10. C
11. FALSE
12. FALSE
13. TRUE
14. TRUE
15. FALSE
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References
Mathematics Learning Module for Grade 10
Teachers Guide
https://en.m.wikipedia.org
https://owlcation.
For inquiries or feedback, please write or call: Department of Education – SOCCSKSARGEN Learning Resource Management System (LRMS)
Regional Center, Brgy. Carpenter Hill, City of Koronadal
Telefax No.: (083) 2288825/ (083) 2281893
Email Address: [email protected]
DISCLAIMER
This Self-learning Module (SLM) was developed by DepEd SOCCSKSARGEN
with the primary objective of preparing for and addressing the new normal.
Contents of this module were based on DepEd’s Most Essential Learning
Competencies (MELC). This is a supplementary material to be used by all
learners of Region XII in all public schools beginning SY 2020-2021. The
process of LR development was observed in the production of this module.
This is version 1.0. We highly encourage feedback, comments, and
recommendations.
