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Mathematics

Quarter 1- Module 3

Illustrating a Geometric Sequence

and Differentiating a Geometric Sequence

from an Arithmetic Sequence

M10AL-Id-1 and M10AL-Id- 2

Mathematics – Grade 10 Quarter 1 – Module 3 Illustrating a Geometric Sequence and Differentiating A Geometric Sequence from an 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 book 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. Published by the Department of Education - Region III Secretary : Leonor M Briones Undersecretary : Diosdado M. San Antonio Department of Education, Schools Division of Bulacan Curriculum Implementation Division Learning Resource Management and Development System (LRMDS) Capitol Compound, Guinhawa St., City of Malolos, Bulacan Mathematics – Grade 10 Supplementary Learning Resource Development Team of the Module Author : Shirley M. Nieto Language Reviewer : Melgee A. Canare Content Editor : Lhio Roem R. Dela Cruz, Ph. D. Illustrator : Shirley M. Nieto Layout Artist : Shirley M. Nieto Management Team Gregorio C. Quinto, Jr., Ed. D. Chief, Curriculum Implementation Division Rainelda M. Blanco, Ph. D. Education Program Supervisor - LRMDS Agnes R. Bernardo, Ph. D. EPS-Division ADM Coordinator Francisco B. Macale Mathematics - Division Focal Person Glenda S. Constantino Project Development Officer II Joannarie C. Gracia Librarian

Introductory Message

For the facilitator: Welcome to the Mathematics 10 Project CAP-LRE Supplementary Learning Resource on Illustrating a Geometric Sequence and Differentiating a Geometric Sequence from an Arithmetic Sequence. This module was collaboratively designed, developed and reviewed by educators from public 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. For the learner: Welcome to the Mathematics 10 Project CAP-LRE Supplementary Learning Resource on Illustrating a Geometric Sequence and Differentiating a Geometric Sequence from Arithmetic Sequence! 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: This will give you an idea of the skills or competencies you are expected to learn in the module. 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. Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners.

This is a brief drill or review to help you link the current lesson with the previous one. In this portion, the new lesson will be introduced to you in various ways; a story, a song, a poem, a problem opener, an activity, or a situation. This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills. 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. This includes questions or blank sentence/paragraph to be filled into process what you learned from the lesson. This section provides an activity which will help you transfer your new knowledge or skill into real life situations or concerns This is a task which aims to evaluate your level of mastery in achieving the learning competency. In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. This contains answers to all activities in the module. At the end of this module you will also find: References This is a list of all sources used in developing this module. 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!

7. What is the 5th^ term of the given geometric sequence 3, 12, 48, …? a. 52 b. 72 c. 192 d. 768
8. Which of the following has a common ratio? a. 6, 12, 18, … b. 6, 12, 24, … c. 6, 10, 14, … d. 6, 18, 30, …
9. Which of the following sequence has a common difference? a. Arithmetic Sequence b. Fibonacci Sequence c. Geometric Sequence d. Harmonic Sequence
10. Common ratio is obtained by what operation? a. Addition b. Subtraction c. Division d. Multiplication
11. Which of the following has a common difference? a. 4, 8, 16, … b. 5, 20, 80, … c. ½, ¼, 1/8, … d. 2, 4, 6, …
12. What is the 7th^ term of the geometric sequence 6, 12, 24, …? a. 192 b. 384 c. 768 d. 1536
13. What is the common ratio of the sequence 5x^3 , 5x^6 , 5x^9 , 5x^12 , …? a. 5x^3 b. 5x^2 c. x^3 d. x^2
14. What is the next term of the given geometric sequence 256, - 128, 64…? a. - 32 b. - 16 c. 16 d. 32
15. Which of the following does not define and illustrate geometric sequence? a. It has a common ratio. b. 5, 25, 125, … c. It is a sequence where the succeeding term is obtained by multiplying the preceding term by a fixed number. d. It is a sequence where the succeeding term is obtained by adding the preceding term by a fixed number. Directions: Find the ratio of the second number to the first number. Express your answer into simplest form.
    1. 2 8
    2. • 5 25
    3. 1 ¼
    4. 24 12
    5. x^3 x^5

Ana and Pepe are putting fresh baked pandesal on the plate. They put 1 pandesal on the first plate, 3 pandesals on the second plate, 9 pandesals on the third plate, 27 pandesals on the fourth plate. If the pattern continues how many pandesals will be on the fifth plate? Observe the sequence Is the given sequence above an example of arithmetic sequence? Let us look at the difference between the terms. It seems that there is no common difference. Thus, this sequence does not belong to an arithmetic sequence. What pattern shall we use to find the number of pandesals which Ana and Pepe will put on the fifth plate? What kind of sequence is it? (^9 27)? 9 - 3 = 27 – 9 =

1 3

3 – 1 =

1 3 9 27

b. 1000, - 100, 10, … Solution: For common ratio

− 100 1000

10 − 100

𝟏 𝟏𝟎 The common ratio is − 𝟏 𝟏𝟎 For next two terms 10 ( − 1 10 )= - 1 and - 1 ( − 1 10

𝟏 𝟏𝟎 The next two terms are - 1 and 1/ Example 2. Write the first 4 terms of a geometric sequence whose first term ( a 1 ) is 7 and common ratio ( r ) is 3. Solution: Given a 1 = 7 and r = 3 Multiply the preceding term by r to get the succeeding term. a 2 = (7)(3) = 21 ; a 3 = (21)(3) = 63 ; a 4 = (63)(3) = 189 The first 4 terms are 7, 21, 63 and 189 The following are the differences of a geometric sequence from an arithmetic sequence: GEOMETRIC SEQUENCE ARITHMETIC SEQUENCE Meaning It is a sequence in which each successive term is obtained from the preceding term by multiplying it by a constant number. It is a sequence in which each successive term is obtained from the preceding term by adding it by a constant number. Constant Number Common Ratio is represented by (r). Common Difference is represented by (d). Operations Involved Multiplication is used to find the next term while Division is to find the common ratio Addition is used to find the next term while Subtraction is to find the common difference Variation of Terms Exponential Linear

Independent Activity 1

Geometric or Not? Directions: Determine whether the given sequence is geometric or not. Put a check mark ( ) before the number if it is geometric and cross mark ( ) if it is not. _______1. 1, 8, 15, 22, …. _______2. 2, 10, 50, 250, … _______3. 2.5, - 10, 40, - 160, … _______4. 90, 30, 10 , 10/3, … _______5. - 1.5, 0, 1.5, 3, …

Independent Assessment 1

Shade Me: True or False Directions: Determine each statement correctly. Shade the circle before each number if the statement is true. Otherwise, leave it blank if the statement is false.

1. Geometric Sequence is a sequence where each term after the first term is obtained by multiplying the preceding term by the common ratio.
2. Common ratio is the fixed number added to each term of the geometric sequence to obtain the next term of the sequence.
3. Arithmetic sequence has a common difference, while geometric sequence has a common ratio.
4. The sequence - 2, 6, - 18, 54 , … has a common ratio.
5. The common ratio of the sequence - 2, - 8, - 32, - 128, … is - 4. In the previous module you learned about arithmetic sequence, where the sequence is obtained by adding the preceding term by the common difference, while in this module you learned about geometric sequence, where the sequence is obtained by multiplying the preceding term by the common ratio.

Independent Activity 3

First 4 Terms Directions: Write the first four terms of the given geometric sequences, using the given first terms and common ratios.

1. a 1 = 5, r = 2 _____, _____, _____, _____
2. a 1 = - 3, r = 5 _____, _____, _____, _____
3. a 1 = ½, r = ½ _____, _____, _____, _____
4. a 1 = 10, r = - 3 _____, _____, _____, _____
5. a 1 = 4, r = 1/3 _____, _____, _____, _____

Independent Assessment 3

Complete Me Directions: Complete the table below. Write whether each of the following sequence is arithmetic or geometric. If it is arithmetic, find the common difference; otherwise find the common ratio if it is geometric. Then write the next term of each sequence. GIVEN ARITHMETIC OR GEOMETRIC COMMON DIFFERENCE (d) COMMON RATIO (r) NEXT TERM

1. 7, 10, 13, 16, …
2. 9, 36, 144 , …
3. 4, 12, 36, 108, …
4. • 6, - 4, - 2, 0, …
5. 2, - 12, 72, - 432 , …
6. 20, 10, 5, 5/2, …
7. 32, 35, 38, 41, …
8. 1/3, 2/3, 1, …
9. 2, 1, ½, ¼, …
10. 17, 12, 7, 2, …

I have learned that Geometric Sequence_____________________________

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The difference between arithmetic sequence and geometric sequence are ________

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A scientist in a laboratory was studying bacteria that can help improve the immune system of the body. If you were a scientist how will you analyze and illustrate this problem. A certain culture of bacteria doubles every three hours. If there are 50 bacteria at 8:00 in the morning, how many bacteria would be there at 11:00 in the morning of the following day?

11. What is the common difference in the arithmetic sequence 0, - 4, - 8, - 12, ...? a. 4 b. 2 c. – 2 d. – 4
12. What is the seventh term of the geometric sequence 2, 4, 8, …? a. 98 b. 108 c. 128 d. 138
13. What is the common ratio of the geometric sequence 1/2500, - 1/500,1/100,-1/20? a. 1/25 b. - 1/5 c. – 5 d. 1/
14. What is the next term in sequence, 2x – 13, 2x - 5, 2x + 3, …? a. 2x + 5 b. 2x + 8 c. 2x + 11 d. 2x + 13
15. Which sequence shows the relationship 𝑎 2 𝑎 1

𝑎 3 𝑎 2

𝑎 4 𝑎 3

a. Arithmetic b. Fibonacci c. Geometric d. Harmonic Illustrate and solve the following problems.

1. An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and so on. How many seats are there in the 8 th^ row?
2. There is a total of 6 bacteria in a dish, and after an hour there is a total of 24 bacteria. After another 3 hours, how many bacteria have just grown in the dish.

References Lopez, Alexander and Recio, Rowell Andrew. Conceptual Math and Beyond 10, K to 12. Reymond Anthony Quan, Teresita Anastacio and Rene Belecina,PhD. Quezon City, Philippines: Brilliant Creations Publishing, Inc. 2014 Tolentino, Joan Eunice, De Gracia, May Maricel, et al. Realistic Math10 Scaling Greater Heights, K to 12. Paulino Gureng.PhD and Socorro Pilor. Quezon City, Philippines: Sibs Publishing House, Inc. 2017 Esparrago, Mirla, Reyes Jr., Nestor and Manalo, Catalina. Next Century Mathematics 10 , K to 12. Jesus Mercado and Fernando Orines. Quezon City, Philippines: Phoenix Publishing House, Inc. 2015 Chu, Tom. Mathematics for the 21st^ Century Learner 10, K to 12. Ignacio Soriano and Joan Michelle Malvas. Makati City, Philippines: DIWA Learning Systems Inc. 2015 Ulpina, Jisela and Razon, Lerida – M. Math Builders 10, K to 12. Josefina Arce. Valenzuela City, Philippines: JO – ES Publishing House, Inc. 2015 Ogena, Ester, Diaz, Rosemarievic and et.al. McGraw–Hill Our Math Grade 10, K to 12. Philippines: The McGraw-Hill Companies, Inc., and Vibal Group, Inc. 2013 AlgebraLAB. “Applications of Sequences and Series”. Accessed July 6, 2020. http://algebralab.org/lessons/lesson.aspx?file=Algebra_SeqSeriesApps.xml. Key Differences. “Difference Between Arithmetic and Geometric Sequence”. Accessed July 8, 2020. https://keydifferences.com/difference-between- arithmetic-and-geometric-sequence.html Purplemath. “Arithmetic & Geometric Sequences”. Accessed July 6, 2 020. https://www.purplemath.com/modules/series3.htm MATH Worksheets 4 Kids. “Geometric Sequence Worksheet”. Accessed July 3, 2020. https://www.mathworksheets4kids.com/geometric-sequence.php

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