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Mathematics

Quarter 1 - Module 2

Arithmetic Sequence and Series

Development Team of the Module Author : Dina Rose G. del Rosario Language Reviewer : Cherrylou C. Dela Cruz Content Editor : Cynthia V. Aguinaldo Illustrator : Mariecris B. Ipapo Layout Artist : Agnes V. Faustino Management Team Gregorio C. Quinto, Jr., EdD Chief, Curriculum Implementation Division Rainelda M. Blanco, PhD Education Program Supervisor - LRMDS Agnes R. Bernardo, PhD EPS-Division ADM Coordinator Glenda S. Constantino Project Development Officer II Francisco B. Macale Mathematics, Division Focal Person Joannarie C. Garcia Librarian II 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 Email address: [email protected] Mathematics - Grade 10 Alternative Delivery Mode Quarter 1 : Arithmetic Sequence and Series First Edition, 2020 Republic Act 8293, section 176 states that; No copyright shall subsist 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

Introductory Message For the facilitator: Welcome to the Mathematics 10 Supplementary Learning Resources on Arithmetic Sequence and Series. 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: 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 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 Supplementary Learning Resources on Arithmetic Sequence and Series. 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 correctly (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.

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! What’s New 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. 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 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 knowledge or skills into real life situations or concerns. Assessment This is a task which aims to evaluate your level of mastery in achieving the learning competency. Additional Activity In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. Answer Key This contains answers to all activities in the module

7. What is the 8th term of the sequence 7, 9, 11, 13, 15, 17,. .? A. 19 B. 21 C. 23 D. 25
8. Which of the following sequences has a common difference of 6? A. 2, 8, 14, 20, … B. 3, 6, 9, 12… C. 5, 15, 20, 25… D. 6, 36, 216…
9. Which of the following is the sum of all the multiples of 3 from 15 to 48? A. 315 B. 360 C. 378 D. 396
10. The first term of an arithmetic sequence is 6 and the common difference is 3. Find the value of the 25th term. A. 75 B. 78 C. 81 D. 84
11. The first term of an arithmetic sequence is 200 and the common difference is - 10. Find the value of the 20th term. A. 0 B. 30 C. 20 D. 10
12. If you are to insert 6 numbers between 3 and 24 to make it an arithmetic sequence, what are the numbers? A. 4, 6, 9, 13, 18, 21 C. 6, 9, 12, 15, 18, 21 B. 5, 8, 10, 13, 15, 20 D. 6, 9, 13, 15, 18, 21
13. From the sequence: 3, 7, 11, ___, 19, ___ …, what are the missing terms? A. 15, 21 B. 15, 22 C. 15, 23 D. 15, 24
14. Find the sum of the first 30 positive even integers. A. 900 B. 910 C. 920 D. 930
15. What is the sum of all odd integers between 20 and 41? A. 341 B. 300 C. 280 D. 261 What’s In An arithmetic sequence is a sequence where every term after the first is obtained by adding a constant called the common difference (d). The sequences 1, 4, 7, 10, ... and 15, 11, 7, 3, ... are examples of arithmetic sequences since each has a common difference of 3 and - 4, respectively. Let us have some exercises on arithmetic sequence.

Exercises: A. Determine the following sequence if arithmetic or not. Write YES if the given illustrates an arithmetic sequence, otherwise write NOT. ___________1. 5, 12, 19, 26, … ___________2. 5, 2, - 1, - 4, … ___________3. 6, 9, 13, 18, … ___________4. 1, 4, 9, 16, … ___________5. - 3, - 9, - 15, - 21, … B. Find the common difference and the next three terms of each arithmetic sequence.

1. 24, 14, 4, ____, ____, ____ d = _____
2. 6, 10, 14, ____, ____, ____ d = _____
3. • 7, 4, 15, ____, ____, ____ d = _____
4. • 8, - 6, - 4, ____, ____, ____ d = _____
5. 4, 11, 18, ____, ____, ____ d = _____ Great! If you answered the exercises by finding the common difference using the formula d = a 2 - a1, where a 1 and a 2 are any two consecutive terms of the sequence, then you did a good job. Let us take the first item in Exercise B, 24, 14, 4, …. The common difference is d = 14 - 24. Hence d = - 10. We obtained the next three terms by adding - 10. a 4 = a 3 + (-10) a 4 = 4 + (-10) a 4 = - 6, Thus, a 5 = - 16 and a 6 = - 26 Now you are ready to explore more about arithmetic sequence. What’s New You already know sequences and their terms. Now, we will look at sequences of numbers that change at a constant rate. Such sequences display arithmetic growth and are useful in modeling many real-life situations. Recall that an arithmetic sequence is a sequence in which each term after the first is found by adding a constant, called common difference d , to the previous term.

Examples: A. Find the 21st^ term of the arithmetic sequence: 4, 7, 10, 13, … Solution: a 1 = 4, d = 3, and n = 21 an= a 1 + (n - 1) d a 21 = 4 + (21 - 1) 3 a 21 = 4 + (20) 3 a 21 = 4 + 60 a 21 = 64 Thus, the 21st^ term is 64. B. Find the 14th^ term of the arithmetic sequence 5, 7, 9, 11, … Solution: a 1 = 5 ; n = 14 ; d = 2 Substitute the values of a 1 , n and d in the formula ; an= a 1 + (n - 1) d a 14 = 5 + (14 - 1) 2 a 14 = 5 + (13) 2 a 14 = 5 + 26 = 31 Thus, the 14th term is 31. Arithmetic extremes Arithmetic means Arithmetic Means The arithmetic mean , also called the average or average value, is the quantity obtained by summing two or more numbers or variables and then dividing the sum by the number of numbers or variables. The arithmetic mean is important in statistics. In the sequence 4, 8, 12, 16, 20, 24,… 4 and 24 - the first and last term of a finite arithmetic sequence are called arithmetic extremes , 8, 12, 16, 20 - the terms between are called arithmetic means. The nth term of an arithmetic sequence is defined as an= a 1 + (n - 1) d where a 1 = the first term an= the nth term d = the common difference Remember:

The arithmetic mean between two numbers is sometimes called the average of two numbers. If more than one arithmetic means will be inserted then employ the formula for d, where Examples: A. What is the arithmetic mean between 10 and 24 10, ______ , 24 using the average formula ; Thus, 17 is the arithmetic mean between 10 and 24. B. Insert three arithmetic means between 8 and 16. 8, _____, _____, _____, 16 a 1 = 8 ; a 5 = 16 Using difference formula: : = = 2 answer Thus 2 is the common difference and is positive. Therefore, add it to the first term to get the succeeding term. Hence, 10, 12, 14, are the arithmetic means. If two nonconsecutive terms of an arithmetic sequence are given, the formula for the common difference is, where an = last given term ak= 1st given term d = the common difference n = position of an k = position of ak Remember: Arithmetic Series A series is the indicated sum of the terms of a sequence. The word “series” can be used both in the singular and in the plural form. An arithmetic series is an indicated sum of the terms of an arithmetic sequence. The sequence : 1, 2, 3, 4, 5, 6, … The series : 1 + 2 + 3 + 4 + 5 + 6, …

Examples: A. Find the sum of the first 100 natural numbers. Solution: The sum of the first 100 natural numbers is 1 + 2 + 3 + ,…+ 100 In this arithmetic series, a 1 = 1, an = 100, n = 100 Since the last term is given use the first formula, Sn= (a 1 + an) Substitute the given, S 100 = (1 + 100) S 100 = 50(101) S 100 = 5050 answer B. Find the sum of the first 14 terms of the arithmetic sequence 2, 5, 8, 11, 15, 17… Solution: The last term is not given so we use the second formula. a 1 = 2, d = 3, n = 14 Sn= [ 2a 1 + ( n – 1)d ] Substitute the given, S 14 = [ 2(2) + ( 14 – 1)3] S 14 = 7 [ 4 + 13(3)] S 14 = 7(43) S 14 = 301 answer C. Find the sum of all multiples of 3 between 1 and 100. Solution: a 1 = 3, an = 99, d = 3 First, we determine how many multiples of 3 there are between 1 and 100. an= a 1 + ( n - 1) d Thus, there are 33 multiples of 3 between 1 and 100. 99 = 3 + (n – 1)3 To find the sum, we use the formula 99 = 3 + 3n - 3 Sn = (a 1 + an) 99 = 3n S 33 = (3 + 99) 33 = n S 33 = 1683 answer

What’s More Independent Activity 1 A. Write the first five terms of the arithmetic sequence from the given data.

1. a 1 = 1; d =
2. a 1 = 15; d = - 3
3. a 1 = - 18; d =
4. a 1 = ; d =
5. a 1 = 2; d = B. Find the specified term of each arithmetic sequence. __________1. 2, 5, 8, … the 8th^ term __________2. 3, 5, 7, … the 21st^ term __________3. 9, 18, 27, … the 10th^ term __________4. 3, 9, 15, 21, … the 50th^ term __________5. 3, 7, 11, 15, … the 25th^ term Independent Assessment 1 Answer the following problems involving arithmetic sequence
6. Find the 16th^ term of the sequence 1, 5, 9, 13, …
7. Find the 11th^ term of the arithmetic sequence 3, 4, 5,…
8. Find the 20th^ term of the arithmetic sequence 17, 13, 9,…
9. If a 1 = 5 and a 7 = 17, find the common difference.
10. In the sequence 50, 45, 40, 35, … , which term is 5? Independent Activity 2 A. What is the arithmetic mean between the two given numbers? ___________1. 5 and 19 ___________2. - 2 and 58 ___________3. 17 and 7 ___________4. 1 and 9 ___________5. - 5 and 3

Independent Assessment 3 DIRECTION: Write the letter of the correct answer.

1. What is the common difference of the Arithmetic Series 14+12+10+8? A. 2 B. – 2 C. 6 D. 44 2.
2. What is the first term or a 1 in finding the sum of the multiples of 5 from 8 to 50? A. 8 B. 50 C. 10 D. 5
3. Love plans to buy her mother a nice gift for Christmas. She intends to start by saving 50 pesos in January, and increasing her savings by 30 pesos each month, thereafter. How much would she have saved by the end of November? A. ₱ 50 B. ₱ 80 C. ₱ 290 D. ₱ 350
4. From problem number 3, how much will Love’s saving in November? A. ₱ 850 B. ₱ 550 C. ₱ 580 D. ₱ 2200
5. Find the sum of the Arithmetic Series if the given are a 1 =7, d=5 and n=15. A. 27 B. 630 C. 525 D. 2625 What I Have Learned Complete the concept of arithmetic sequence by choosing the correct word inside the parenthesis. I have learned in this module that an arithmetic sequence is a list of numbers where the _____ (1. sum , difference) between each number is the same or constant. The constant number is called the _____ (2. term , common difference). To tell whether or not a sequence is arithmetic, we need to be able to see what is happening between each _____ (3. term , constant). Understanding arithmetic sequences, and how to identify them is a great way to develop critical thinking skills. For example, in this sequence, 5, 12, 19, 26…, we can tell that this is an arithmetic sequences by _____ (4. adding , subtracting) each term from the one following it. Meanwhile, arithmetic _____ (5. mean , series) is a value that is between any two terms of the sequence. Hence _____ (6. 5 and 26, 12 and 19) are the arithmetic means in the example. An arithmetic series is the _____ (7. sum , difference) of the terms in an arithmetic sequence with or without a definite number of terms. The formula for the sum if the last term is given is ____(8. , , ) .When the last term is not given, the formula to find the sum is (9. , )

What I Can Do It is alarming that many people now are being infected by COVID-19. As a student in your school, you invited people to give a five-day series of talks on COVID-19 and its prevention every first Friday of the month from 12 noon to 1 P.M. using any online platform. On the first day, 20 students attended. Finding the talk interesting, these 20 students shared the talk to other students and 10 more students attended on the second day, another 10 more students on the third day, and so on. a. Assuming that the number of participants continues to increase in the same manner, make a table representing the number of participants from day 1 of the talk until day 5. b. Represent the data in the table using a formula. Use the formula to justify your data in the table. c. You feel that there is still a need to extend the series of talks, so you decided to continue it for three more days. If the pattern continues where there are 10 additional students for each talk, how many students are there in all? Assessment DIRECTION: Write the letter of the correct answer.

1. Find the 25th term of the arithmetic sequence 3, 7, 11, 15, 19,... a. 109 b. 99 c. 90 d. 89
2. Give the common difference of the arithmetic sequence – 15, - 10, - 5, … a. 5 b. 10 c. – 5 d. – 10
3. Which term of the arithmetic sequence is - 18, given that a 1 = 7 and a 2 = 2? a. 6th term b. 5th term c. 4th term d. 3rd term
4. Which term of the arithmetic sequence 18, 11, 4, …. Is - 73? a. 24th term b.11th term c. 21st term d. 14th term
5. Find the twelfth term of 16, 12, 8, …. a. 28 b. 26 c. – 28 d. – 26

Answer Key What I Know D 1. B 2. A 3. B 4. D 5. C 6. B 7. A 8. C 9. B 10. D 11. C 12. C 13. D 14. B 15. Exercises A. YES 1. YES 2. NOT 3. NOT 4. YES 5. B. 10 - d = 26 - 16,- 6,– 1. d = 4 18, 22, 26 2. d = 11 26, 37, 48 3. d = 2 2, 0, 2- 4. d = 7 25, 32, 39 5. What’s More 1 Independent Activity A. 1, 2, 3, 4, 5 1. 15, 12, 9, 6, 3 2. 4, 3, 10- 11,- 18,- 3. 1/6, 2/3, 7/6, 5/3, 13/6 4. 2, 5, 8, 11, 14 5. B. 23 1. 43 2. 90 3. 297 4. 99 5. Independent Assessment 1 61 1. 13 2. 59 - 3. 2 4. 10th 5. Independent Activity 2 A. 12 1. 28 2. 12 3. 5 4. 1 - 5. B. 8 1. 49, 67 2. 11, 14, 17 3. Independent Assessment 2 A 1. D 2. B 3. B 4. C 5. Practice 1 5 + 10 + 15 + 20 + 25 … 1. 15 … - 12 - 9 - 6 - 3 - 2. 7 … - 2 - 8 + 3 3. 7 + 2 +11 + 20 …- 4. 1 + 5 + 9 + 13 … 5. Practice 2 2, 5, 8, … 1. 3, 5, 7, … 2. 6, 0, …- 12,- 3. 5, 8, 11, … 4. 12, 21, 30, ... 5. Independent Activity 3 100 1. 1023 2. 275 3. 310 4. Independent Assessment 3 B 1. C 2. D 3. D 4. B 5. Assessment B 1. A 2. A 3. D 4. C 5. B 6. A 7. C 8. C 9. D 10. D 11. D 12. A 13. D 14. B 15. Additional Activity Presentation of the answer may vary. Answer: 9th What I Have Learned difference 1. Common difference 2. term 3. subtracting 4. mean 5. 12 and 19 6. Sum 7.

What I Can Do A. B. Presentation of answer may vary [2(20) + 7(10)] =^8 S C. = 4 (40 + 70)^8 S = 440^8 S 5th 4th 3rd 2nd 1st 60 50 40 30 20

References

Jose-Dilao, Soledad, Intermediate Algebra for Second Year , Revised Edidtion, Julieta G. Bernabe_._ Quezon City : SD Publications, Inc. 2009 Callanta, Melvin M., Canonigo Allan M., Chua Arnaldo I., Cruz Jerry D., Esparrago Mirla S., Garcia, Elino S., Magnaye, Aries N., Orines Fernando B., Perez Rowena S., and Ternida, Concepcion S., Learning Materials for Grade 10 , First Edition, Pasig City: REX Book Store. 2015 Mallari, Ma. Theresa G., Sabangan, Leticia G., Gabriel, Judy, Villareal, Abelardo A. Argel, Analyn M., Semira,, Gemmalyn T., Angeles, Alma, Gestoso, Bernie A., Math Time for Grade 10, Quezon City: Educational Resources Corporation. 2015 Floresca Jr., Emeterio Jovinal, Math for Today’s Learner 10 , Lucena: Kampana Publishing House, Inc. 2014