10

NAME:__________________________________________
GRADE/SECTION:______________________________

MATHEMATICS
Quarter I – Week 3
Difference Between Geometric
Sequence and Arithmetic Sequence

CONTEXTUALIZED LEARNING ACTIVITY SHEETS

SCHOOLS DIVISION OF PUERTO PRINCESA CITY
Mathematics – Grade 10
Contextualized Learning Activity Sheets (CLAS)
Quarter I - Week 3: Difference Between Geometric Sequence and Arithmetic Sequence
First Edition, 2021

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Development Team of the Contextualized Learning Activity Sheets

Writer: R-Ley John M. Gacis

Content Editors: Haydee C. Hitosis and Francisco D. Pańa

Language Editors: Eden E. Cardaño PhD and Norman Anthony C. Java

Proofreader: Joeffrey A. Padual

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Illustrators: Randy N. Corpuz and Cresen Gay N. Bacosa

Layout Artists: Jezreel D. Abellanosa and Eryl G. Gamuyao

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Telephone No.: (048) 434 9438
Email Address: puertoprincesa@deped.gov.ph
 Let’s Try
Directions: Read and analyze each item. Write the letter of the correct answer on the
space provided before each number.

_____1. What is the next term in the geometric sequence 2, -4, 8?

A. -16 B. -12 C. 12 D. 16

_____2. In a geometric sequence 3, 9, 27, what is the common ratio?

A. 2 B. 3 C. 4 D. 6

_____3. What is a sequence where each term after the first is obtained by multiplying the
preceding term by a nonzero constant?
A. arithmetic sequence C. geometric sequence
B. harmonic sequence D. fibonacci sequence

_____4. Which of the following is a geometric sequence?

A. 16, 12, 8, 4 B. 3, 6, 9, 12 C. 4, 8, 16, 32 D. 1, 2, 3, 4

_____5. What is the next term in the geometric sequence 5, 15, 45?
A. 135 B. 125 C. 75 D. 160

_____6. Which of the following is NOT a geometric sequence?

A. 64, 16, 4, 1 C. 4, 24, 144
B. 5, 5, 5, 5 D. 8, 12, 16, 20

_____7. If common difference is for arithmetic sequence, what is for geometric sequence?
A. common denominator C. common factor
B. common product D. common ratio

_____8. What is the common ratio of the geometric sequence 16, 8, 4, 2?

1
A.4 B. 2 C. D. -2
2

_____9. What is the next term in the geometric sequence -3, -6, -12?
A. -24 B. -16 C. -4 D. -2

_____10. What is a constant number multiplied to each term of a geometric sequence to

obtain the next term of the sequence?
A. common denominator C. common factor
B. common product D. common ratio

1
 Lesson 1
Geometric Sequence

MELC: The learners should be able to illustrate a geometric sequence. (M10AL-Id-1)

Objectives: 1. Define a geometric sequence.

2. Identify the common ratio of a geometric sequence.
3. Illustrate a geometric sequence.

Let’s Explore and Discover

Mr. John plans to buy a car five years from now. This year, he
Unlocking
canvass the price of the car that he wants to buy. The price of
of Difficulties
the car is Php 2,000,000 which depreciates by 5% at the end of
each year.
• Sequence is a
particular order in
which related
events or things
follow each other.

• Ratio is a
relationship
between two
quantities,
expressed as the
quotient of one
divided by the
other. C.G. Bacosa

Questions:
a. What is the price of the car on the fifth year?
_____________________________________________________________________________
b. How does the geometric sequence facilitate in finding solutions to real-life
problems?
______________________________________________________________________________
_____________________________________________________________________________

To answer these questions, you should read and understand this topic. Are you
ready? Let us begin.

2
Geometric Sequence

Geometric Sequence is a sequence where each term after the first is obtained by
multiplying the preceding term by a nonzero constant called the common ratio.

An example of a geometric sequence is 1, 3, 9, 27 …

Step 1:
In the given sequence, each term is multiplied by 3 to get the next term.

1, 3, 9, 27 …

x3 x3 x3
Step 2:
Thus, the next two terms are 81 and 243.

1, 3, 9, 27, 81, 243 …

x3 x3
Step 3:
The common ratio can be determined by dividing any term in the sequence by the
term precedes it or to find the common ratio, divide the second term by the first, or the
third by the second, or the fourth by the third, and so on. Therefore, the common ratio
of the given sequence is 3.

𝟑 𝟗 𝟐𝟕 𝟖𝟏
= 𝟑, = 𝟑, = 3, = 3, …
𝟏 𝟑 𝟗 𝟐𝟕

Note that the letter r is used to denote the common ratio.

Example 1

Find the common ratio and the next two terms of each geometric sequence.

a. 4, 8, 16, 32,... b. 54, 18, 6, 2,…

Solution:
a. 4, 8, 16, 32,...

Step 1:
4, 8, 16, 32 …

x2 x2 x2
Step 2:

4, 8, 16, 32, 64, 128 …

x2 x2

3
Step 3:

𝟖 𝟏𝟔 𝟑𝟐
r= = 𝟐, 𝐨𝐫 = 𝟐, 𝐨𝐫 = 2, …
𝟒 𝟖 𝟏𝟔

Answer:
Therefore, the common ratio (r) is 2 and the next two terms are 64 and 128.

b. 54, 18, 6, 2,…

Step 1:
54, 18, 6, 2 …
𝟏 𝟏 𝟏
x𝟑 x𝟑 x𝟑
Step 2:

𝟐 𝟐
54, 18, 6, 2, ,𝟗…
𝟑

𝟏 𝟏
x𝟑 x𝟑
Step 3:

𝟏𝟖 𝟏 𝟔 𝟏 𝟐 𝟏
r= = , or = , 𝐨𝐫 = ,…
𝟓𝟒 𝟑 𝟏𝟖 𝟑 𝟔 𝟑

Answer:
𝟏 2 2
Therefore, the common ratio (r) is and the next two terms are and .
𝟑 3 9

• In general, a geometric sequence is in the form

𝑎1, 𝑎1 𝑟, 𝑎1 𝑟 2 , 𝑎1 𝑟 3 , … , 𝑎1 𝑟 𝑛−1 ,
where 𝑎1 is the first term, n is the number of terms, and r is the common ratio.

(Source: Melvin M. Callanta et al., Mathematics 10: Learner’s Module,

Pasig City: DepEd-IMCS, 2015, 27.)

4
 Let’s Practice

ACTIVITY 1 Directions: Identify the ratio of the second number to the first
number. Write your answer on the space before each number.

_________1. 3, 12 _________6. 1, 6
_________2. 4, -24 _________7. 6, 3
_________3. 8, 2 _________8. -8, 8
_________4. -5, -15 _________9. 20, 100
1
_________5. 2, _________10. 7, 7
2

ACTIVITY 2 Directions: Find the next missing term in each geometric

sequence.

1. 3, 9, 27, ___ 6. 500, 100, 20, ___

2. 1, 5, 25, ___ 7. 5, -10, 20, ___
3. 11, 22, 44, ___ 8. -2, -8, -32, ___
1 1 1
4. 4, 12, 36, ___ 9. 81 , 27 , 9 , ___
5. -3, 3, -3, ___ 10. 2, 2, 2, ___

How do you determine the next missing term of the geometric sequence?
_______________________________________________________________________________________

Let’s Do More

ACTIVITY 1 Directions: Solve the given problem involving the geometric

sequence. Write your answer on the space provided.

Problem:

Ma’am Elsa wants to decorate her classroom wall with butterfly paper folding. To
maximize the use of a colored paper, she cut it in half. Each half is again cut in half.
If this process is repeated 5 times, what will be the area of each piece of paper?

5
 ACTIVITY 2 Directions: State whether the following sequence below is
geometric or not.

Sequences: Geometric or Not Geometric

1. 3, 12, 48,…

2. 5, 10, 15, 20,…

3. 2, -14, 98, -686,…

4. -3, 0, 0, 0, 0,…

5. 16, 8, 4, 2, …

How do you identify that a sequence is a geometric sequence?

_______________________________________________________________________________________

Let’s Sum It Up
DIRECTIONS Fill in the blanks. Choose your answer from the box below.

arithmetic sequence n multiplying

common ratio t dividing

geometric sequence 𝑎𝑛 common difference

r 𝑎1

1. A _________________ is a sequence where each term after the first is obtained by

multiplying the preceding term by a nonzero constant called the
_______________.
2. The common ratio can be determined by ________ any term in the sequence by
the term precedes it.
3. The letter __ is used to denote the common ratio.
4. In general, a geometric sequence is in the form
2 3
𝑎1, 𝑎1 𝑟, 𝑎1 𝑟 , 𝑎1 𝑟 , … , 𝑎1 𝑟 𝑛−1 , where ____ is the first term, ____ is the number of
terms, and r is the common ratio.

6
 Lesson 2
Difference Between Geometric
Sequenceand Arithmetic Sequence
MELC: The learners should be able to differentiate a geometric sequence from an arithmetic
sequence. (M10AL-Id-2)

Objective: 1. Recall arithmetic and geometric sequences and their common difference and
common ratio respectively.
2. Determine whether the given sequence is arithmetic or geometric in nature.
3. Find the next terms of arithmetic and geometric sequences.
4. Differentiate a geometric sequence from an arithmetic sequence.

Let’s Explore and Discover

Roger is being pushed on a swing by his friend, reaching a

maximum height of 4 feet. His friend stops pushing, and the
Unlocking maximum height of the swing decreases by 10% on each
of Difficulties successive swing.

• Geometric
Sequence is a
sequence where
each term after
the first is
obtained by
multiplying the
preceding term by
a nonzero
constant called
the common ratio.

C.G. Bacosa

Questions:
a. What is the maximum height of the swing during the 5th swing?
_____________________________________________________________________________
b. Is this situation shows arithmetic or geometric sequence?
______________________________________________________________________________
_____________________________________________________________________________

How does the difference between the arithemetic and geometric sequence facilitate
in finding solutions to real-life problems?

7
To answer these questions, you should read and understand this topic. Are you ready? Let
us begin.

If arithmetic sequences are formed by addition, geometric sequences are formed by

multiplication. Each term in a geometric sequence is obtained by multiplying the previous
term by the common ratio, while each term in an arithmetic sequence is obtained by
adding the previous term by the common difference.

An example of a geometric sequence is 1, 5, 25, 125, … where 5 is the common

ratio. On the other hand, an example of an arithmetic sequence is 1, 5, 9, 13, … where
4 is the common difference.

Also, be reminded that common ratio of a geometric sequence is denoted by letter

r. For the common difference of an arithmetic sequence, it is denoted by letter d.

Examples:

Determine whether each sequence is arithmetic or geometric, then find the next 3 terms.

a. 12, 14, 16, 18,… b. 5, 10, 15, 20,…

c. 32, 16, 8, 4,… d. 1, 2, 4, 8,…

Solution:

a. 12, 14, 16, 18,…

• Find out if there is a common difference.

d = 𝑎2 − 𝑎1 = 14 – 12 = 2

d = 𝑎3 − 𝑎2 = 16 – 14 = 2

d = 𝑎4 − 𝑎3 = 18 – 16 = 2

The common difference is 2. Therefore, the sequence is an arithmetic.

• Find out if there is a common ratio.

𝑎2 14 7
r= = =
𝑎1 12 6

𝑎3 16 8
r= = =
𝑎2 14 7

𝑎4 18 9
r= = =
𝑎3 16 8

There is no common ratio. Therefore, the sequence is not geometric.

8
 Since, the sequence 12, 14, 16, 18,… is an arithmetic sequence and its common
difference is 2, then the next 3 terms are 20, 22, and 24.

b. 5, 10, 15, 20,…

• Find out if there is a common difference.

d = 𝑎2 − 𝑎1 = 10 – 5 = 5

d = 𝑎3 − 𝑎2 = 15 – 10 = 5

d = 𝑎4 − 𝑎3 = 20 – 15 = 5

The common difference is 5. Therefore, the sequence is an arithmetic.

• Find out if there is a common ratio.

𝑎2 10
r= = =2
𝑎1 5

𝑎3 15 3
r= = =
𝑎2 10 2

𝑎4 20 4
r= = =
𝑎3 15 3

There is no common ratio. Therefore, the sequence is not geometric.

Since, the sequence 5, 10, 15, 20,… is an arithmetic sequence and its common
difference is 5, then the next 3 terms are 25, 30, and 35.

c. 32, 16, 8, 4,…

• Find out if there is a common difference.

d = 𝑎2 − 𝑎1 = 16 – 32 = - 16

d = 𝑎3 − 𝑎2 = 8 – 16 = - 8

d = 𝑎 4 − 𝑎3 = 4 – 8 = - 4

The is no common difference. Therefore, the sequence is not arithmetic.

9
 • Find out if there is a common ratio.

𝑎2 16 1
r= = =
𝑎1 32 2

𝑎3 8 1
r= = =
𝑎2 16 2

𝑎4 4 1
r= = =
𝑎3 8 2

1
The common ratio is . Therefore, the sequence is a geometric.
2

Since, the sequence 32, 16, 8, 4,… is an geometric sequence and its common ratio
1 1
is , then the next 3 terms are 2, 1, and .
2 2

d. 1, 2, 4, 8,…

• Find out if there is a common difference.

d = 𝑎2 − 𝑎1 = 2 – 1 = 1

d = 𝑎 3 − 𝑎2 = 4 – 2 = 2

d = 𝑎 4 − 𝑎3 = 8 – 4 = 4

The is no common difference. Therefore, the sequence is not arithmetic.

• Find out if there is a common ratio.

𝑎2 2
r= = =2
𝑎1 1

𝑎3 4
r= = =2
𝑎2 2

𝑎4 8
r= = =2
𝑎3 4

The common ratio is 2. Therefore, the sequence is a geometric.

Since, the sequence 1, 2, 4, 8,… is an geometric sequence and its common ratio is
2, then the next 3 terms are 16, 32, and 64.

(Source: Melvin M. Callanta et al., Mathematics 10: Learner’s Module,

Pasig City: DepEd-IMCS, 2015, 37.)

10
 Let’s Practice

ACTIVITY 1 Directions: Find the common ratio following geometric

sequences. Write your answer on the space before each number.

_________1. - 5, - 10, - 20, - 40

_________2. 16, 8, 4, 2

_________3. 2, 6, 18, 54

_________4. 7, 7, 7, 7

_________5. -1, 1, -1, 1

ACTIVITY 2 Directions: Find the next missing terms in each sequence. Fill out
the table.

Arithmetic or
Sequences: Missing terms
Geometric

1. -81, 27, -9, __, __

2. -12, -6, 0, __, __

1
3. , 1, 7, __, __
7

4. 8, 8, 8, __, __

5. 7, 4, 1, __,__

How do you determine the ratio of the geometric sequence?

_______________________________________________________________________________________
_______________________________________________________________________________________

11
 Let’s Do More

ACTIVITY 1 Directions: Solve the given problem involving the geometric

sequence. Write your answer on the space provided.

Problem:

Your younger brother is playing a ball inside his bedroom. While dribbling the ball, it
slips from his hand and accidentally, the ball bounces out through the window. Each
time it hits the ground, it bounces to 80% of the previous height. Give the geometric
sequence of the first five height of the ball’s bounce, if the window is 12 ft. above the
ground.

ACTIVITY 2 Directions: Find the next missing terms in each sequence. Fill out
the table.

Arithmetic common
Sequences: or difference / Missing terms
Geometric common ratio

1. 11, 22, 44, __, __, __

2. 49, 40, 31, __, __, __

3. 2, - 6, 18, __, __, __

How do you whether is it a common ratio or a common difference in a sequence?

_______________________________________________________________________________________
______________________________________________________________________________________

12
 Let’s Sum It Up

DIRECTIONS Differentiate arithmetic and geometric sequences using a two-

column chart.

Arithmetic Sequence Geometric Sequence

1.

2.

3.

13
 Let’s Assess
Directions: Read and analyze each item. Write the letter of the correct answer on the
space provided before each number.

_____1. What is the common ratio of the geometric sequence 16, 8, 4, 2?

1
A. -2 B. C. 2 D. 4
2

_____2. What is the next term in the geometric sequence -3, -6, -12?
A. -2 B. -4 C. -16 D. -24

_____3. What is a sequence where each term after the first is obtained by multiplying the
preceding term by a nonzero constant?
A. arithmetic sequence C. geometric sequence
B. harmonic sequence D. fibonacci sequence

_____4. Which of the following is NOT a geometric sequence?

A. 64, 16, 4, 1 C. 4, 24, 144
B. 5, 5, 5, 5 D. 8, 12, 16, 20

_____5. What is the next term in the geometric sequence 2, -4, 8?

A. 16 B. 12 C. -12 D. -16

_____6.In a geometric sequence 3, 9, 27, what is the common ratio?

A. 6 B. 4 C. 3 D. 2

_____7. If common difference is for arithmetic sequence, what is for geometric sequence?
A. common denominator C. common factor
B. common product D. common ratio

_____8. What is a constant number multiplied to each term of a geometric sequence to

obtain the next term of the sequence?
A. common denominator C. common factor
B. common product D. common ratio

_____9. Which of the following is a geometric sequence?

A. 16, 12, 8, 4 B. 3, 6, 9, 12 C. 4, 8, 16, 32 D. 1, 2, 3, 4

_____10. What is the next term in the geometric sequence 5, 15, 45?
A. 60 B. 75 C. 125 D. 135

14
 Answer Key
Let’s Let’s Practice (LESSON 1) Let’s Do More Let’s Do More
Try Activity 1: Activity 2: Activity 1: Activity 2:
1. 4 1. 81 1
Answer : 32 1. Geometric
1. A. 2. -6 2. 125 2. Not Geometric
2. B. 3. 4
1
3. 88 3. Geometric
3. C. 4. Not Geometric
4. 3 4. 108
4. C. 1 5. Geometric
5. A. 5. 4 5. 3
6. D. 6. 6 6. 4
7. D. 7.
1
7. -40 Let’s Sum It Up
8. C.
2
1. geometric sequence, common ratio
8. -1 8. -128
9. A. 1 2. dividing
9. 5 9. 3 3. r
10. D.
10. 1. 10. 2 4. 𝑎1 , n

Let’s Practice (LESSON 2) Let’s Do More

Activity 1: Activity 2: Activity 1:
1
1. 2 1. Geometric ; 3, The geometric sequence of the first five height of
3
1 the ball’s bounce is 12, 9.6, 7.68, 6.14, 4.92, 3.93.
2. 2 2. Arithmetic ; 6, 12
Activity 2:
3. 3 3. Geometric ; 49, 343 1. Geometric ; 2 ; 88, 176, 352
4. 1 4. Geometric ; 8, 8 2. Arithmetic ; 9 ; 22, 13, 4
5. -1 5. Arithmetic ; -2, -5 3. Geometric ; -3 ; - 54, 162, -486

Let’s Sum It Up Let’s Assess

1. common difference ; common ratio 1. B. 4. D. 7. D.
2. d ; r 2. D. 5. D. 8. D.
3. formed by addition ; formed by multiplication 3. C. 6. C. 9. C.
10. D.

References

Module

Callanta, Melvin M., Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz, Mirla S. Esparrago,
Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, Rowena S. Perez, and
Concepcion S. Ternida. Mathematics 10 Learner’s Module.Pasig City, Philippines:
Department of Education, 2015.

Gacis, R-Ley John M., Haydee C. Hitosis, Francisco D. Paña, Eden E. Cardaño PhD,
Norman Anthony C. Java, Marie Vic C. Velasco PhD, Joeffrey A. Padual, Juliet C.
Medez, Randy N. Corpuz, Cresen Gay N. Bacosa, Jezreel D. Abellanosa, and Eryl G.
Gamuyao. Mathematics 10 Module 9. Puerto Princesa City Division: Department of
Education Contextualized Learning-Instruction Kit (CLIK), 2020.

15
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