Philippines Mathematics Grade 10 Quarter 1 Module 2 Arithmetic Sequence 2019 PDF

10 Mathematics Quarter I - Module 2: Illustrating an Arithmetic Sequence Mathematics – Grade 10 Alternative Delivery Mode Quarter I – Module 2: Illustrating an Arithmetic Sequence First Edition, 2019 Republic Act 8293, section 176 states that: No copyright s...

10 Mathematics Quarter I - Module 2: Illustrating an Arithmetic Sequence Mathematics – Grade 10 Alternative Delivery Mode Quarter I – Module 2: Illustrating an Arithmetic Sequence First Edition, 2019 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 Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Development Team of the Module Writer’s Name: Evelyn N. Ballatong Editor’s Name: Heather G. Banagui Reviewers’ Name: Bryan A. Hidalgo, Laila B. Kiw-isen, Jim D. Alberto, and Selalyn B. Maguilao Management Team: May B. Eclar, PhD, CESO V, Regional Director Benedicta B. Gamatero, ASDS, OIC-SDO Ifugao Carmel F. Meris, CES-CLMD Marciana M. Aydinan, PhD, Chief- CID, SDO Ifugao Ethielyn E. Taqued, RO EPS/LRMDS Manager Edgar H. Madlaing, RO EPS for AP/ADM Focal Person Lydia I. Belingon, Division EPS-Math/ADM Focal Person Printed in the Philippines by: Department of Education – Cordillera Administrative Region Office Address: Wangal, La Trinidad, Benguet Telefax: (074) 422-4074 E-mail Address: [email protected]  10 Mathematics Quarter 1 - Module 2 Illustrating an Arithmetic Sequence M10AL – Ib – 1 and M10AL-If-2 Introductory Message This module was collaboratively designed, developed and reviewed by educators from both 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. For the facilitator: This module deals with the second learning competency in our Mathematics 10 curriculum standards; hence mastery of the skills is significant to have a smooth progress in the succeeding lessons. This learning material also designed to equip the students with essential knowledge about defining and illustrating arithmetic sequences. With this, please be patient and encourage the learner to complete this module. For the learner: This module deals with the second learning competency in our Mathematics 10 curriculum standards; hence mastery of the skills is significant for you to have a smooth progress in the succeeding lessons. This learning material serves as a bridge from the previous lesson to the next lesson. By doing the prepared activities, it is expected from you to define and illustrate arithmetic sequence. Please read completely the written texts and follow the instructions carefully so that you will be able to get the most of this learning material. We hope that you will enjoy learning. 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 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 What I Know about the lesson to take. If you get all the answers correct (100%), you may decide to skip this module. This is a brief drill or review to help What’s In you link the current lesson with the previous one.  In this portion, the new lesson will be introduced to you in various ways What’s New such as 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 What Is It help you discover and understand new concepts and skills. This comprises activities for independent practice to solidify your understanding and skills of the topic. What’s More 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 filed in to What I have Learned process what you learned from the lesson. This section provides an activity which will help you transfer your new What I Can Do knowledge or skill into real life situations or concerns. This is a task which aims to evaluate Assessment your level of mastery in achieving the learning competency. In this portion, another activity will be given to you to enrich your knowledge Additional Activities or skill of the lesson learned. This also tends retention of learned concepts. This contains answers to all activities Answer Key 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 though this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it!  What I Need to Know This module was designed and written with you in mind. It is here to help you define and illustrate an arithmetic sequence. 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 pacing in which you read and answer this module is dependent on your ability. After going through this module, you are expected to: 1. define an arithmetic sequence, 2. illustrate an arithmetic sequence and 3. solves problems involving arithmetic sequence. What I Know Find out how much you already know about the topics in this module. Choose the letter of the best answer. Take note of the items that you were not able to answer correctly and find the right answer as you go through this module. Write the letter of your choice on a separate sheet of paper. 1. Which of the following is a correct description of an arithmetic sequence? a. A term is obtained by multiplying a constant number from the preceding term. b. A term is obtained by squaring the preceding term. c. A term is obtained by adding a constant number to the preceding term. d. A term is obtained by extracting the square root of the preceding term. 2. Which of the following is the common difference of the arithmetic sequence 4, 7, 10, 13, …? a. 0 b. 1 c. 2 d. 3 3. What is the common difference in the sequence: -4, 3, 10, 17,, …? a. 1 b. -7 c. 7 d. -1 4. Which of the following is the missing term of the arithmetic sequence, -1, -7, _____, -19, -25, …? a. -10 b. -11 c. -12 d. -13 5. Which of the following of the common difference of the arithmetic sequence -1, -8, -15, -22, …? a. -5, b. -6 c. -7 d. -8 6. Which of the following is the common difference in the arithmetic sequence: 1 , 1, 1 1 , 2, …? 2 2 1 3 5 1 a. b. c. d. 4 4 2 2 7. Which of the following is an arithmetic sequence? a. -7, 4, 15,26,... c. 21, 15, 9, 2,... b. -8, -6, -4, 0, … d. 4, -1, -7, -13,... 8. Which of the following is the common difference of an arithmetic sequence whose 𝑎2 = 1, 𝑎5 = 7? a. 1 b. 2 c. 3 d. 4 9. Which of the following is the common difference of the arithmetic sequence 3a -1, 3a, 3a + 1, … ? a. -a b. a c. -1 d. 1 10. Which of the following is the first positive term of the arithmetic sequence -8, -6, -4, …? a. 0 b.2 c.4 d. 6 11. Which of the following is the value of a to make 3a +1, 4a, 6a + 1, … an arithmetic sequence? a. -2 b. -1 c. 0 d. 1 12. Which of the following is not an arithmetic sequence? a. 2, 4, 6, 8, … c. 3, 6, 12, 24,… b. 4.1, 11.1, 18.1,… d. 5, 10, 15, 20, … 13. Which of the following is not an arithmetic sequence? a. 4, 7, 10, 13, … c. 100, 98, 96, 94, … b. -2, -6, -10, -14, … d. -4,-3, 10, 17, … 14. Which of the following is not true about the arithmetic sequence 5, 3, 1, -1, -3, … a. 𝑎2 = -3 b. the common difference is -2 c. 𝑎6 = -5 d. one way to obtain a term, -2 must be added to the preceding term. 15. I was advised by my physician to walk each day in the morning as my daily exercise. On my first day, I walked 40m. On the second and third day, I walked 60m and 80m, respectively, and so on. Which of the following is the distance I walked on the 10th day if I continue the pattern in my daily walk? a. 180 m b. 200 m c. 220 m d. 240 m Lesson Illustrates an arithmetic sequence. 1 What’s In In the previous module, you have learned that a sequence is an arrangement of objects, numbers or even figures which follows a certain pattern. Also, you have learned about the processes in finding patterns of any sequence. Look at the sequences below. Can you see the specific pattern they follow? 2, 4, 6, 8, … 3, 6, 12, 24, … 5, 10, 15, 20, … Let us try to give emphasis on the differences you observed as we proceed with this module. What’s New Look at the following sequences. What is the pattern you observed in each sequence? Sequences Pattern 1. 2, 4, 6, 8, … A term is obtained by adding 2 to the preceding term. 2. 3, 6, 12, 24, … A term is obtained by multiplying 2 to the preceding term. 3. 5, 10, 15, 20, … A term is obtained by adding 5 to the preceding term.  What is It Observe the following sequences: 1. 4, 7, 10, 13, … 2. 33, 38, 43, 48, … 3. -2, -6, -10, -14, … 4. 100, 98, 96, 94, … 5. 1 , 1, 1 1 , 2, … 2 2 Can you give the next two terms of the above sequences? How did you get those terms? If you get 16 and 19 in item 1, then you are correct. Notice that a constant number 3 is added to the preceding term to get the next term. To get the next term in item 2, 5 is added to the preceding term. To get the next term in items 1 3, 4 and 5, the numbers -4, -2, and 2 are added to the preceding term, respectively. Notice that to get the next term in each of the sequences above, a constant or a common number is added to the preceding term or the number before it. The constant number being added is called the common difference and we represent it as d. All these sequences are called arithmetic sequences. To find the common difference (d), you can simply subtract the second term (𝑎2 ) by the first term (𝑎1 ), 𝑎2 - 𝑎1 , or the third term (𝑎3 )by the second term (𝑎2 ), 𝑎3 - 𝑎2 , or the fourth term ( 𝑎4 ) by the third term (𝑎3 ), 𝑎4 - 𝑎3 , or in general, a term (𝑎𝑛 ) by its preceding term (𝑎𝑛−1 ) 𝑑 = 𝑎𝑛 − 𝑎𝑛−1. Arithmetic Sequences A sequence in which term after the first is formed by adding a fixed number to the preceding term is called arithmetic sequence. The fixed number or constant is called the common difference denoted by 𝒅.  What’s More Activity 1: From the discussion on arithmetic sequence earlier, solve the problems that follows. 1. Determine if the sequence is arithmetic or not. If it is, find the common difference and the next three terms of the sequence. -4, 3, 10, 17, … Solution: a. To find out if the sequence is arithmetic, there must be a common difference between any two consecutive terms in the sequence. 𝑎2 - 𝑎1 = 3 – (-4) =7 𝑎3 - 𝑎2 = 10 – (3) =7 𝑎4 - 𝑎3 = 17 – 10 =7 Because there is a common difference between consecutive terms, the sequence is arithmetic. b. Since the common difference is 7, the next three terms are obtained by adding 7 to the preceding term. 𝑎5 = 𝑎4 + 7 = 17 + 7 = 24 𝑎6 = 𝑎5 + 7 = 24 + 7 = 31 𝑎7 = 𝑎6 + 7 = 31 + 7 = 38 Thus, the common difference is 7 and the next three terms are 24, 31, 38. 2. Write the first five terms of the arithmetic sequence with 5 as the first term and with a common difference of -2. Solution: First term: 𝑎1 = 5 Second term: 𝑎2 = 𝑎1 + (-2) =5–2 =3  Third term: 𝑎3 = 𝑎2 + (-2) =3–2 =1 Fourth term: 𝑎4 = 𝑎3 + (-2) =1–2 = -1 Fifth term: 𝑎5 = 𝑎4 + (-2) = -1 – 2 = -3 To get a term, we added the common difference or the constant to the preceding term. So, the first five terms of the sequence are 5, 3, 1, -1, and -3. 3.Find the common difference in an arithmetic sequence whose 𝑎2 = 1, 𝑎5 = 7. Solution: a. Remember that to find a term we add the common difference 𝑑, to the preceding term. For instance, 𝑎3 = 𝑎2 + 𝑑 Equation 1 𝑎4= 𝑎3 + 𝑑 Equation 2 𝑎5 = 𝑎4 + 𝑑 Equation 3. b. Substitute 𝑎3 in Equation 1 to Equation 2 𝑎4= 𝑎3 + 𝑑 𝑎4= (𝑎2 + 𝑑) + 𝑑 𝑎4= 𝑎2 + 2𝑑 Equation 4 c. Substitute 𝑎4 in Equation 4 to Equation 3 𝑎5 = 𝑎4 + 𝑑 𝑎5 = (𝑎2 + 2𝑑) + 𝑑 𝑎5 = 𝑎2 + 3𝑑 Equation 5 d. Substitute the given in Equation 5 𝑎5 = 𝑎2 + 3𝑑 7 = 1 + 3𝑑 Equation 6 e. Solve for 𝑑 in Equation 6. 7 = 1 + 3𝑑 6 = 3𝑑 2= 𝑑 So, the common difference, 𝑑 is 2. 4. Find the common difference of the arithmetic sequence 3a -1, 3a, 3a + 1, … Solution: a. We must remember that in an arithmetic sequence, the common difference 𝑑 is a term minus the preceding term. Thus 𝑑 = 𝑎2 − 𝑎1 Equation 1 or 𝑑 = 𝑎3 − 𝑎2 Equation 2  b. If we use Equation 1 and substitute the given, we have 𝑑 = 3a – (3a -1) c. If we simplify, we obtain 𝑑 = 1. So, the common difference is 1. 5. Find the value of a to make 3a +1, 4a, 6a + 1, … an arithmetic sequence? Solution: a. We must again remember that in an arithmetic sequence, the common difference 𝑑 is a term minus the preceding term. Thus 𝑑 = 𝑎2 − 𝑎1 Equation 1 and 𝑑 = 𝑎3 − 𝑎2 Equation 2 b. If we substitute the given in Equation 1 and Equation 2, 𝑑 = 4a – (3a + 1) Equation 3 𝑑 = (6a + 1) – 4a Equation 4 c. Equate Equation 3 and Equation 4 4a – (3a + 1) = (6a + 1) – 4a d. Combine similar terms a – 1 = 2a + 1 e. Further simplify -2 = a Thus, the value of a is -2. 6. I was advised by my physician to walk each day in the morning as a daily exercise. On the first day, I walked 40m. On the second and third day, I walked 60m and 80m, respectively, and so on. Which of the following is the distance I walked on the 10 th day if I continue the pattern in my daily walk? Solution: a. If we analyze the problem, 40m, 60m, 80m, … follows an arithmetic sequence since there is a common difference which is equivalent to 20. Dropping the unit, the given could be written as follows 40 , 60, 80, … b. Continuing this pattern until the 10th term, we have 40 , 60, 80, 100, 120, 140, 160, 180, 200, 220, … Thus, in the 10th day, the distance travelled is 220 m. Activity: Find the common difference and the next three terms of each arithmetic sequence. Write your answer on your answer sheet. Common Difference Next 3 Terms _________ 1. 24, 14, 4, ___, ___, ___ _________ 2. 6, 10, 14, ___, ___, ___ _________ 3. -7, 4, 15, ___, ___, ____ _________ 4. 21, 15, 9, ___, ___, ___ _________ 5. -8, -6, -4, ___, ___, ___ _________ 6. 5, -1, -7, ___, ___, ___ _________ 7. 4.1, 11.1, 18.1, ___, ___, ___ _________ 8. -1, -8, -15, ___, ___, ___ _________ 9. -3x, -10x, -17x ___, ___, ___ _________ 10. 3a -1, 3a, 3a + 1, ______, ______, ______ What I Have Learned Answer the questions that follows on your answer sheet. a. What is an arithmetic sequence? b. How do we get the next terms of an arithmetic sequence? What I Can Do There is Math Around Us Arithmetic sequence can be observed around us. Like the following fare rate for first 4 kms of a modernized PUJ under General Community Quarantine released by LTFRB last April 24, 2020. Distance Fare First kilometer 11.00 Second kilometer 12.50 Third kilometer 14.00 Fourth kilometer 15.50 If we compute the increase of fare for every increase of kilometer distance, they are all equivalent to 1.50. With this, the fare rate is an example of an arithmetic sequence. Aside from examples involving money, identify three situations or three things that you see or observe in your surroundings that illustrate an arithmetic sequence. Write your answer in your answer sheet. Assessment Choose the letter that you think best answers the question. Write your answer on your answer sheet. 1. Which of the following describes an arithmetic sequence? a. A sequence in which a term is formed by adding any number to the preceding term. b. A sequence in which there is an equal difference between consecutive terms. c. A sequence in which a term minus the preceding term is always positive. d. A sequence in which terms follows a pattern. 2. Which of the following is the common difference in the sequence: 0, 4, 8, 12, …? a. 1 b. 2 c. 3 d.4 3. Which of the following is the common difference in the sequence: 3, -2, -7, …? a.1 b. -5 c. 5 d. -1 4. Which of the missing term in this arithmetic sequence: 23, 18, 13, 8, 3, _____, -7, -12,… ? a.-2 b. 2 c. -5 d. 5 5. Which of the following is the common difference in the sequence: -7, -4, -1, 2, 5,…? a.-3 b. 3 c. 4 d. -4 6.Which of the following is the common difference in the arithmetic 13 7 15 sequence: 3, 4 , 2, 4 , … ? 1 3 5 a. b. c. d. 4 4 4 2 7. Which of the following is an arithmetic sequence? a.1, 2, 3, 5, 7, 9… c. 1, -1, -3, -5… b.1, 10, 20, 30… d. 7, -7, 7, -7 …  8. Which of the following is the common difference of an arithmetic sequence if 𝑎3 = 4 and 𝑎5 = 14? a.6 b. 5 c. 4 d. 3 9. Which of the following is the common difference of the arithmetic sequence 7p + 2, 5p +12, 3p + 22, …? a.2p b. -2p +10 c. 2p - 10 d. 6p 10. Which of the following is the first positive term of the arithmetic sequence: -11, -8, -5, …? a. -4 b. 3 c. -2 d. 1 11. Which of the following is the value of p so that the terms, 7p + 2, 5p +12, 2p – 1,… form an arithmetic sequence? a. -8 b. -5 c. -13 d. -23 12. Which of the following is NOT an arithmetic sequence? a. -5, -2, 1, 4 c. 1, 4, 7 b. 11, 14, 17, 20 d. 3, 7, 12, 18 13. Which of the following is not an arithmetic sequence? a.1, 2, 3, 4, 5, … c. 4.5, 5.0, 5.5, 6.0, … b.3, 9, 27, 81, … d. 13, 2, -9, -20, -31, … 14. Which is NOT true about the arithmetic sequence: 25, 32, 39, 46, …? a. The common difference is 7. c. the 6th term is 60. b. The 7th term is 60. d. the 8th term is 74. 15. During a free-fall rappelling, a skydiver jumps 36 feet, 48 feet, and 60 feet on the first, second, and third fall, respectively. If he continues to jump at this pattern, how many feet will he have jumped during the tenth fall? a.144 b. 156 c. 132 d. 140 Additional Activity Determine whether the following situations illustrate an arithmetic sequence. If yes, then give what is asked. Write your answer on your answer sheet. 1. I was advised by my physician to walk each day in the morning as my daily exercise. On my first day, I walked 40m. On the second and third day, I walked 60m and 80m, respectively, and so on. What is the distance I walked on the 10th day if I continue the pattern in my daily walk? 2. The number of works done by a backhoe in a certain area doubles every 2 hours. If there are N number of works to start with, find the number of works done in 14 hours.  Assessment: 1. A 2. D 3. B 4. A 5. B 6. A 7. C Additional Activity 8. B 9. B 1. YES- 10. D In my 10th day of 11. D walking, I would What I Can Do 12. D have walked 220m. Depends on student’s 13. B 14. B responses. 15. A 2. No 16. What’s More 1.d= -10 -4,-14, -24 What I Have Learned 2.d= 4 18, 22, 26 What I Know 1.Arithmetic sequence - 3.d= 11 26, 37, 48 1. C A sequence in which term 4.d=-6 3, -3, -9 2. D 3. C after the first is formed by 5.d=2 -2, 0, 2 4. D 5. C adding a fixed number to 6. D 6.d=-6 -13, -19, -25 7. A the preceding term. 8. C 7.d= 7 25.1, 32.1, 39.1 9. D 2. By adding a fixed 10. B 8.d= -7 -22, -29, -36 11. A number or constant from 12. C 9.d= -7x -24x,-31x,-38x 13. D the preceding term. 14. A 10.d= 1 3a+2, 3a+3, 3a+4 15. D Answer Key References Melvin M. Callanta, et al. Mathematics Learner’s Module. Pasig City, 2015. Gladys C. Nivera and Minie Rose C. Lapinid, Grade 10 Mathematics: Patterns and Practicalities. Makati City, Don Bosco Press, 2015. Land Transportation Franchising and Regulatory Board. Guidelines for Public Transportation for Areas Under General Community Quarantine.2020. http://ltfrb.gov.ph/wp-content/uploads/2020/04/MC-2020-017-RE- GUIDELINES-FOR-PUBLIC-TRANSPORTATION-AREAS-UNDER-GCQ- min.pdf For inquiries or feedback, please write or call: Department of Education - Bureau of Learning Resources (DepEd-BLR) Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600 Telefax: (632) 8634-1072; 8634-1054; 8631-4985 Email Address: [email protected] * [email protected]: