10

Mathematics
Quarter 1 – Module 5 : Geometric
Sequence vs. Arithmetic
Sequence

Department of Education Republic of the Philippines

Mathematics – Grade 10
Alternative Delivery Mode
Quarter 1 – Module 6: Differentiates a geometric sequence from an arithmetic
sequence. - M10AL-Id-2
First Edition, 2019

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 10
Mathematics
Quarter 1 – Module 5:
Differentiates a Geometric
Sequence from an Arithmetic
Sequence
M10AL-Id-2

This instructional material was collaboratively developed and

reviewed by educators from public and private schools, colleges,
and/or universities. We encourage teachers and other education
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We value your feedback and recommendations.

Department of Education Republic of the Philippines

1
Introductory Message
For the facilitator:

As the facilitator of this module, please be patient in guiding the learner as he/she
undergo this module. Always remind the learner the value of patience and
perseverance. Do not forget to remind the learner to use separate sheets in
answering all of the activities found in this module.

For the learner:

I hope that this module will be of big help in completing your Mathematics 10. This
is designed to provide you with interactive tasks to further develop the desired
learning competencies prescribed in the K-12 Curriculum of the Department of
Education. After undergoing this module, you are expected to accurately
differentiate a geometric sequence from an arithmetic sequence.

Here is a guide on the parts of the learning modules which you need to understand
as you progress in reading and analyzing its content.

ICON LABEL DETAIL

What I need to know This contains the learning objectives
which you need to accomplish
What I know This assesses what you know about
the lesson you are to tackle
What’s In This connects the current lessons
with the previous lessons
What’s New This introduces the lesson through an
activity
What Is It This contains a brief discussion of the
lessons
What’s More These are activities to check your
understanding of the lesson
What I have Learned This summarizes the important ideas
presented in the lesson
What I Can Do This is a real life application of what
you have learned
Assessment This is a post assessment of what you
have learned
Additional Activities This is an activity that will strengthen
your knowledge about the lesson

What I need to 2
Know
 This module was designed and written with you in mind. It is here
to help you differentiate arithmetic sequence from geometric sequence.
The scope of this module permits it to be used in many different learning
situations. The lessons are arranged to follow the standard sequence of
the course but the pacing in which you read and answer this module will
depend on your ability.

After going through this module, the learner should be able to:
a. determine whether a sequence is geometric or arithmetic.
b. explain inductively the difference between arithmetic sequence
and geometric sequence using real life-situations.

What I Know
Multiple Choice. Read and analyze the following items and determine the
letter of the correct answer from the given choices. Write your answer on a
separate sheet of paper.
_____ 1. What operation is usually used in obtaining the next term in a
geometric sequence?
A. Addition C. Division
B. Multiplication D. Subtraction
_____ 2. What operation is usually used in obtaining the next terms in an
arithmetic sequence?
A. Addition C. Division
B. Multiplication D. Subtraction
_____ 3. Which of the following sequences is arithmetic?
25 27
A. 12, - 12, - 12, -12 C. 12, , 13,
2 2
1
B. 32, 8, 2, D. 1, 2, 3, 5, 8
2
_____ 4. Which of the following sequences is geometric?
25 27
A. 12, - 12, - 12, -12 C. 12, , 13,
2 2

3
 1
B. 32, 8, 2, D. 1, 2, 3, 5, 8
2
_____ 5. Which term is closely related to a geometric sequence?
A. Common Ratio C. Both A & B
B. Common Difference D. None of these
For Items 6 – 10, determine whether the given situation is arithmetic,
geometric or neither. Write A for arithmetic, G for geometric and N for
neither.
_____ 6. An amount of money which have a compound interest rate of 6 %
annually
_____ 7. Saving 5 pesos on your piggy bank daily.
_____ 8. Pushing a swing with a constant force.

_____ 9. Each minute hand of the clock that the second hand covers is 5
seconds.

_____ 10. Half – life of a carbon or any element.

_____ 11. Growth of bacteria on a petri dish.
_____ 12. Distance covered by a runner who constantly increases his speed/
minute.
_____ 13. Graph of a linear function.
_____ 14. A car running with a constant speed.
_____ 15. Your heartbeat at rest per minute.

4
 Lesson
Arithmetic Sequences vs.
1 Geometric Sequences

It was discussed to you earlier that sequences are arrangement of

objects or terms which follow a certain pattern. Look around you, can you
cite situations, figures, or even routines which follow a certain pattern?
What’s In
Before going to school, are their sequenced routines that you follow? When
performing something, do you follow sequential actions?
These are just situations where we can see the applications of
sequences.
Every individual has his or her own choices and decisions. Each follows
a different sequence or a pattern as a result of his choices and decisions.
Just like us, humans, different sets of numbers follow different sequences
or patterns.

What’s New

HOW TO GENERATE SEQUENCES?

Determine the pattern being followed by the terms or numbers in each of
the following sequences.
1. 1, 2, 3, 4, 5, … ______________________
2. 7, 14, 28, 56, … ______________________
3
3. 12, 6, 3, , … ______________________
2
4. 7, 12, 17, 22, … ______________________
5. 13, 7, 1, - 5, … ______________________
If you are going to observe the given sequences, there is always a
constant value that is being added or multiplied to obtain the terms in
the sequence.
Let us discover these concepts through this module.

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 What is It

Look at the two sequences below, what can you say about the pattern
that is being followed by the terms or numbers in each sequence? What is
the pattern you observed in Sequence 1? How about in Sequence 2?

1 5, 12, 19, 26, 33, … 2 3, 12, 48, 192,…

In the first example, 5 is the first term or denoted as a1, then followed by
the second term (a2) 12, third term (a3 ) 19, and so on. You can observe that
in order to obtain the second term (a2), you add 7 to the first term (a1), and
to obtain the third term (a3), you add 7 to the second term. So, the pattern is
7 is being added to the preceding terms to obtain the next terms. The
number that is being added is referred to as the common difference(d).

+7
5+7, 12+7, 19+7, 26 , 33, … Common difference

The common difference can be

obtained by :
a 2−a1or a 2−a1 or a 2−a1 or
Remember
Succeeding term – preceding

An arithmetic sequence is a sequence obtained by

adding a common difference (d) to the preceding term in
order to obtain the next term

In the second example, 3 is the first term and you can see that 12 is
the second term (a2) this was obtained by multiplying 4 to a1. Multiplying 4
again to a2 will give you the third term (a3 ), which is 48, and so on. So, the
pattern is 4 is being multiplied to the preceding terms to obtain the
next terms. The number that is being multiplied is the common ratio.

Common
x4
3x4, 12x4, 48 , 192,…
The common ratio can be
obtained by :
a2 a3 a4
a1
or a2
or a3
6
succeeding term
preceding term
 Remember

A geometric sequence is a sequence obtained by multiplying a common

ratio to the preceding terms in order to obtain the succeeding terms.

What’s More
ACTIVITY 1: ARITHMETIC OR GEOMETRIC!
Determine whether the sequence is geometric or arithmetic.
1. 12, 15, 18, 21,…
2. 10, 5, 0, - 5
3. 5, 15, 45,…
4. 24, 12, 6, 3,…
Example 1.) 12, 15, 18, 21,…
First, let’s investigate if a common ratio or a common difference
exist. From the first three terms: 12, 15, and 18.
a2 15 5 a3 18 6
= = and = =
a1 12 4 a2 15 5

We got the ratio of two consecutive terms but, we failed to

obtain a common ratio. So, the sequence is not a geometric sequence.
Now, let’s try to obtain a common difference by getting the first
three terms 12, 15, 18.

a2 – a1 = 15 – 12 = 3, then a3 – a2 = 18 – 15 = 3
We performed subtraction of two consecutive terms and, we
were able to obtain a common difference. Therefore, we can conclude
that the sequence is arithmetic.

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 Let us do the same for examples 2 to 4 to find out if the sequence is
arithmetic or geometric.

Example 2.) 10, 5, 0, -5,…

a2 5 1 a3 0 ½ is not equal to 0. Thus, sequence is
=
10
=
2
and =
5
=0
not geometric
a1 a2

There is a common difference of – 5.

a2 – a1 = 5 – 10 = - 5 and a3 – a2 = 0 – 5 = - 5 Therefore, the sequence is
arithmetic

Example 3.) 5, 15, 45,…

There is a common difference of 3.
a2 15 a3 45
= = 3 and = =3 Therefore, the sequence is geometric
a1 5 a2 15

Example 4.) 24, 12, 6, 3,…

1
a2 12 1 a3 6 1 There is a common difference of .
2
= = and = =
a1 24 2 a2 12 2 Therefore, the sequence is geometric

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ASSESSMENT 1. WHICH IS WHICH?
Directions: Determine if the following sequence is arithmetic, geometric or
neither. Write JUST for arithmetic, DO for geometric and IT for neither.
Write your answer on the box above the sequence.
1

2.) – 1 , 1, – 1 , 1 ,…
1.) a, ar, ar2, ar3,… 4.) 40, 20, 10, 5,…

3.) 1, 1, 2, 3, 5, 8,…

6.) x, x + y , x + 2y , x + 3y,…
1 2 3 4
5.) , , , ,
2 3 4 5
… 8.) 1, 3, 5, 7,…

3
7.) – 3, – , 0,…
2
10.) 3, 6, 12, 24,…

9.) 2, 6, 18, 54, 162,…

ASSESSMENT 2. THEY NEED HELP

Instructions: Read and analyze the situation carefully. Then, provide what is
being asked.
SITUATION:
Your classmates Ruben and Nathaniel are arguing during your Math
subject. You found out that they are arguing about the examples of
sequences presented by your teacher.

B. An amount of money deposited in a

A. Saving 5 pesos on your
1
bank that is compounded yearly at an
piggy bank daily.
interest rate of 6 %.
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 Ruben is telling that example A is an arithmetic sequence, while
example B is a geometric sequence. On the other hand, Nathaniel is
defending that example A is a geometric sequence, while example B is an
arithmetic sequence.
Since you are knowledgeable about the issue, who among your
classmates will you agree with?
Answer: ___________
What explanation/s will you provide to your classmates for them to
understand clearly the difference between an arithmetic sequence and
geometric sequence?

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
_____________________________________________________________________

Rubrics:
4 3 2 1
-Argument is -Argument is -Argument gives -Argument gives
very informative, somewhat some new no new
accurate and informative and information but information and
well organized. organized. poorly organized. poorly organized.

Congratulations! You did it right!

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 What I have
learned
SEE THE DIFFERENCE!
What is the main difference that you observed between the two
sequences below?

4 4 4 27 29
A. 12, 4, , , … B. 13, , 14, , 15 …
3 9 27 2 2

___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
____________________________.

What I can do

INVESTIGATE AND CONCLUDE

Read and analyze the situation below.
Consider yourself depositing an amount of money in a local bank
which offers 5% interest rate compounded annually. If you deposited
P10,000 on that bank, how much will you earn after 3 years?
1. What kind of sequence is illustrated in the above situation?

2. If it shows an arithmetic sequence, then what is the common difference?

On the other hand, if it shows a geometric sequence, then what is the

11
 common ratio? Provide a justification for your answer using any method.

Assessment
Read and analyze the following items and determine the letter of the correct
answer from the given choices. Write your answer on a separate sheet of
paper.
_____ 1. What kind of sequence is 7, 20, 33, …?
A. Arithmetic C. Harmonic
B. Geometric D. Fibonacci
7
_____ 2. 14, 7, … is an example of ____________ sequence.
2
A. Arithmetic C. Harmonic
B. Geometric D. Fibonacci
For items 3 – 5, refer to the situation below.
This Christmas season, Cesar is planning to buy a new mobile phone
worth P10, 000.00. He started saving P200 pesos on the first month of the
year and on the second month, he increases his savings by half of his initial
saving.
_____ 3. If he continues to follow the same pattern in the succeeding
months, what will be his total savings at the end of the year?
A. P7, 688 . 67 C. P18, 700 . 00
B. P17, 299 . 51 D. P11, 533 . 00
_____ 4. What is the common ratio of the sequence obtained from his
savings?
1 3
A. C.
2 2
2 1
B. D.
3 3
_____ 5. When will he be capable of buying the mobile phone?
A. at the end of September C. at the end of November
B. at the end of August D. at the end of October

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For Items 6 – 15, determine whether the given situation is arithmetic,
geometric or neither. Write A for arithmetic, G for geometric and N for
neither.
_____ 6. Half – life of a carbon or any element
_____ 7. An amount of money which have a compound interest rate of 10 %.
_____ 8. Saving a money on your piggy bank 10 pesos weekly.

_____ 9. Each minute hand of the clock that the second hand covers is
5seconds.

_____ 10. Graph of a linear function.

_____ 11. Your heartbeat at rest per minute.
_____ 12. Distance covered by a runner who constantly increases his speed
per minute.
_____ 13. Growth of bacteria on a petri dish.
_____ 14. A car running with a constant speed.
_____ 15. Pushing a swing with a constant force.

Job Well Done!

REFERENCES
www.VectorStock.com/15088776
www.yumpu.com/en/document/read/35995428
Cabral,Josephine,Bernabe,Julieta,and Valencia,Efren.New Trends in Math
Series,Algebra II. Araneta Avenue,Quezon City: Vibal Publishing House
Incorporated, 2004.

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