M10-q1-M3-Arithmetic-Sequence-v2.pdf

10 Mathematics Quarter I - Module 3: Arithmetic Means and 𝒏𝒕𝒉 term of an Arithmetic Sequence Mathematics – Grade 10 Alternative Delivery Mode Quarter I – Module 3: Arithmetic Means and 𝒏𝒕𝒉 term of an Arithmetic Sequence First Edition, 2019 COPYRIGHT PAGE 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 Reviewer’s Name: Bryan A. Hidalgo, Laila B. Kiw-isen, Jim D. Alberto, Selalyn B Maguilao Management Team: May B. Eclar Benedicta B. Gamatero Carmel F. Meris Marciana M. Aydinan Ethielyn E. Taqued Edgar H. Madlaing Lydia I. Belingon 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] ii 10 Mathematics Quarter I - Module 3: Arithmetic Means and 𝒏𝒕𝒉 term of an Arithmetic Sequence M10AL – Ib – c – 1 and M10AL-If-2 Introductory Message This module was collaboratively designed, developed and reviewed by educators both from public and private institution to assist you, the teacher or facilitator in helping the leaners meet the standards set by the K to 12 Curriculum while overcoming their personal, social and economic constraints in schooling. This module deals with the third learning competency in our Mathematics 10 curriculum standards; hence mastery of the skills is significant to have smooth progress in the next lesson. This learning materials is also designed to equip the students with essential knowledge about finding the 𝑛𝑡ℎ term and arithmetic means of an arithmetic sequence. For the facilitator: Hi. As the facilitator of this module, kindly orient the learner on how to go about in reading and answering this learning material. Please be patient and encourage the learner to complete this module. By the way, do not forget to remind the learner to use separate sheets in answering all of the activities found in this module. For the learner: Hello learner. I hope you are ready to progress in your Grade 10 Mathematics by accomplishing this learning module. This is designed to provide you with interactive tasks to further develop the desired learning competencies on determining arithmetic means and 𝑛𝑡ℎ term of an arithmetic sequence. This module is especially crafted for you to be able to cope with the current lessons taken by your classmates. 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 about What I Know 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 you What’s In link the current lesson with the previous one. 2 In this portion, the new lesson will be introduced to you in various ways such What’s New 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. 3 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! 4 What I Need to Know This module was designed and written with you in mind. This will help you determine arithmetic means and 𝑛𝑡ℎ term of an arithmetic sequence. The scope of this module will 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 how you read and answer this module is dependent on your ability. After going through this module, you are expected to be able to demonstrate knowledge and skill related to sequence and apply these in solving problems. Specifically, you should be able to: 1. write a formula for the 𝑛𝑡ℎ term of an arithmetic sequence, 2. find the 𝑛𝑡ℎ term or unknown term of an arithmetic sequence, 3. define arithmetic means, 4. determine arithmetic means of a sequence and 5. solves problems involving 𝑛𝑡ℎ term of an arithmetic sequence. 5 What I Know Find out how much you already know about the topics in this module. Choose the letter of your answer from the given options. Take note of the items that you are not able to answer correctly and find the right answer as you go through this module. Write your answer on a separate sheet of paper. 1. Which term of the arithmetic sequence 7, 10, 13, 16,... is 304? a. 99th term b. 100th term c. 111th term d. 102th term 2. Find the nth term of the arithmetic sequence given the following conditions: 𝑎1 =2 d= 3 n=9 a. 26 b. 27 c. 28 d. 29 3. Which term of the arithmetic sequence 2, 6, 10,….. is 102? a. 20th term b. 26th term c. 30th term d. 35th term 4. If three arithmetic means are inserted between -15 and 9, find the first of these arithmetic means. a. 3 b. -3 c. -6 d. -9 5. Find the 21st term of the arithmetic sequence 6, 9, 12, 15,… a. 61 b. 60 c. 62 d. 66 6. If three arithmetic means are inserted between 8 and 16, find the second arithmetic mean. a. 10 b. 12 c. 14 d. 16 7 5 7. Which term of the arithmetic sequence 3, 3, 3, …, … is -27? a. 9th term b. 20th term c. 41th term d. 46th term 8. What is the arithmetic mean between 10 and 24? a. 18.5 b. 19 c. 16 d. 17 9. What is the 10th term of the following arithmetic sequence: -5, -1, 3, 7, 11,…? a. 31 b. 19 c. 27 d. 22 10. Insert two arithmetic means between √2 and 4 √2. Which of the following is the first arithmetic mean? a. √2 b. 2√2 c. 3√2 d. 4√2 11. If a1 = -4 and a25 = -100. Find a100? a. -104 b. -150 c. -316 d. -400 12. If a3 = 8 and a16 = 47 and ak is the kth term of the sequence and ak = 212, then what is the value of k? a. 61 b. 71 c. 81 d. 91 13. Insert 2 arithmetic means between 3 and 30. a. 12, 14 b. 12, 11 c. 12, 21 d. 12, 30 6 14. After one second, a rocket is 30 ft above the ground. After another second, it is 85 feet above the ground. Then after another second, it is already 140 feet above the ground. If it continues to rise at this rate, how many feet above the ground will the rocket be after 16 seconds? a. 780 ft b. 830 ft c. 855 ft d. 910 ft 15. An object is dropped from a plane and falls 32 feet during the first second. For each succeeding second, it falls 40 feet more than the distance covered in the preceding second. How far has it fallen after 11 seconds? a. 118 feet b. 220 feet c. 315 feet d. 432feet 7 Lesson Finding the 𝒏𝒕𝒉 Term of an 1 Arithmetic Sequence What’s In In the previous module, we define what an arithmetic sequence and find the next term of a sequence by adding a constant number. For example: Find the next three terms of the arithmetic sequence: 3, 8, 13, 18, … Solution: a. The terms are a1 = 3, a2 = 8, a3 = 13, and a4 = 18. So, we will be finding a5, a6, and a7. b. The common difference (d) in the sequence is 5. c. To get the next three terms, add 5 to each of the preceding term. Thus: a5 = a 4 + 5 = 18 + 5 = 23 a6 = a 5 + 5 = 23 + 5 = 28 a7 = a 6 + 5 = 28 + 5 = 33 What about if we are required to the 100th term or the 250th term, given the 1st term and the common difference? How can we find the terms? Using the process that is illustrated above will take much of our time and effort. There is a short cut in doing this and that is one of the foci of this module. 8 What’s New Before we find other higher terms of a sequence, let us first find lower terms. In the arithmetic sequence: 3, 8, 13, 18,…; what is the 15th term? Solution: a. By adding the common difference to each of the preceding terms, we get the following values. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 an 3 8 13 18 23 28 33 38 43 48 53 58 63 68 73 b. Thus, the 15th term is 73. However, using this procedure to get any higher n th term would be tedious. Thus, a formula is necessary to find any nth term. What is It Let us investigate on how to determine the nth term of a sequence. In the table: a1 = 3 =3 a2 = 3 + 5 =8 a3 = 3 + 5 + 5 = 13 a4 = 3 + 5 + 5 + 5 = 18...... a13 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 63 a14 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 68 a15 = 3 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 73 These terms can be written in the following manner as a short cut. a1 = 3 =3 a2 = 3 + 5 (1) =8 a3 = 3 + 5 (2) = 13 a4 = 3 + 5 (3) = 18...... a13 = 3 + 5 (12) = 63 a14 = 3 + 5 (13) = 68 a15 = 3 + 5 (14) = 73 9 Thus, if we find for the 16th term of the arithmetic sequence, then a16 = 3 + 5 (15) = 78. 3 is the first term 5 is the common difference 15 is one less the number of term. With these, we can derive that pattern that the nth term of the sequence is an = a1 + d (n-1), where: an is the term that corresponds to nth position, a1 is the first term, and d is the common difference. The nth term of an arithmetic sequence with first term a1 and common difference d is given by: an = a1 + d (n-1) What’s More Let us apply the formula in solving the following problems: A. Find the 21st term of the arithmetic sequence: 6, 9, 12, 15,… Solution: a. From the sequence, 𝑎1 = 6 , d = 3, and n = 21. b. Using the formula, substitute these values. a21 = 6 + 3 (21 – 1) a21 = 6 + 3 (20) a21 = 6 + 60 a21 = 66 c. Thus, the 21st term is 66. B. In the arithmetic sequence: 7, 10, 13, 16,...; find n if an = 304. Solution: a. From the sequence, a1 = 7, d = 3, and an = 304. b. Using the formula, substitute these values. an = a1 + d (n-1) 304 = 7 + 3 (n – 1) 304 = 7 + 3n – 3 304 = 4 + 3n 300 = 3n n = 100 10 c. Thus, 304 is the 100th term of the sequence. C. The 3rd term of an arithmetic sequence is 8 and the 16th term is 47. Find d, 𝑎1 and the 71st term. Solution: a. From the sequence, a3 = 8 and a16 = 47 b. These imply that: a3 = a1 + d (3-1) a16 = a1 + d (16-1) 8 = a1 + d (3-1) 47 = a1 + d (16-1) 8 = a1 + 2d Eq. 1 47 = a1 + 15d Eq. 2 c. Using Eq. 1 and Eq. 2, solve for a1 and d. By subtracting Eq. 2 by Eq. 1, then: 47 = a1 + 15d – (8 = a1 + 2d) 39 = 13d d=3 To solve for a1, substitute d = 3 to either Eq. 1 or Eq. 2. Using Eq. 1: 8 = a1 + 2(3) 8 = a1 + 6 a1 = 2 Thus, the nth term of the arithmetic sequence is an = 2 + 3(n-1) d. Using an = 2 + 3(n-1), we can solve for the 71st term. a71 = 2 + 3(71-1) a71 = 2 + 3(70) a71 = 2 + 210 a71 = 212 Alternative Solution: Another way to solve d is to use the 𝑎𝑛 −𝑎𝑘 difference formula: 𝑑 = 𝑛−𝑘 Given : ak = a3 = 8; k = 3 and an = a16 = 47; n = 16 a −a Thus, d = n k n−k 47 −8 = 16−3 39 = 13 =3 D. After one second, a rocket is 30 ft above the ground. After another second, it is 85 feet above the ground. Then after another second, it is already 140 feet above the ground. If it continues to rise at this rate, how many feet above the ground will the rocket be after 16 seconds? Solution: a. From the problem we let the given be a1 = 30 a2 = 85 a3 = 140 11 b. Find first d by substituting the given value of a1 and a2 in the formula then simplify. an = a1 + d (n-1) 85= 30 + d (2-1) 55 = d c. To find a16 , the unknown in the problem substitute the obtained value of d and the given value of a1 in the formula then simplify. a16 = a1 + d (16-1) = 30 + 55 (16-1) = 855 d. Thus, the rocket will b 855 ft above the ground after 16 seconds. Activity: Write answer on your answer sheet. A. Find the specified nth term of each arithmetic sequence. ___________1. 2, 5, 8, …; 9th term ___________ 2. 3, 5 7, …; 20th term 1 ___________ 3. 1, , 0, …; 16th term 2 ___________ 4. 5, 11, 17, …; 9th term ___________ 5. 26, 22, 18, …; 40th term ____________6. 103rd term of the arithmetic sequence if 𝑎1 = -5 and d = -4 ____________7. 19th term of the arithmetic sequence if 𝑎1 = 25 and d = -2 1 3 ____________8. 25th term of the arithmetic sequence if 𝑎1 = and d = −. 2 8 B. Solve what is asked. ____________ 1. In the sequence 2, 6, 10, …; find n if the nth term is 102. 7 5 ____________ 2. In the sequence 3, , , …; find n if the nth term is -27. 3 3 ____________ 3. Find the15th term of the sequence if 𝑎8 = 5 and 𝑎21 = -60 ____________ 4. Find 5th term of the sequence if 𝑎15 = 29 and 𝑎27 = 47 ____________ 5. If a1 = −4 , a25 = −100 , what is the value of a100 ? 12 What I Have Learned Let us see if you understood our lesson by answering the following problems. Write answer on your answer sheet. 1. What is the general formula of finding the nth term of an arithmetic sequence? 2. Given an arithmetic sequence, how do we find the common difference? 3. Given two different nth terms of an arithmetic sequence, how do we find for the common difference? What I Can Do A. Give what is asked: 1. The 10th term of the arithmetic sequence if 𝑎1 = -15 and d = 6 1 2. The 39th term of the arithmetic sequence if 𝑎1 = 40 and d = 2 B. Find the specified term of each arithmetic sequence. 1. 1.4, 4.5, 7.6, …; the 41st term 2. 9, 18, 27,…; the 23rd term 3. 14, 6, -2,…; 27th term 4. 3, 3.25, 3.5,…; 16th term 5. 1, 4, 7,… ; 28th term C. Find the specified term. 1. In the sequence: 0.12, 0.17, 0.22, …; find n if the nth term is 0.67? 2. In the sequence: 10, 7, 4, …; what term has a value of -296? 3. In the sequence: 2, 6, 10, 14, …; what n corresponds to an= 286? 4. Find 1st term of the sequence if a5 = 26 and a12 = 47. 5. If a24 = 85 , and a28 = 100 , what is a1 ? 13 Lesson Computing Arithmetic Means 2 What’s In In the previous lesson, you learned how to determine the nth term of an arithmetic sequence. For example: In the sequence: 10, 15, 20, 25,…; what term has a value of 385? Solution: a. Using the formula, an = a1 + d(n – 1): 385 = 10 + 5 ( n – 1 ) 385 = 10 + 5n -5 385 = 5n + 5 5n = 385 – 5 5n = 380 n = 76 b. Thus, 385 is the 76th term of the given sequence. The next lesson intends to discuss with you how to compute arithmetic means. What’s New The focus of this part of the module has something to do with finding the arithmetic means. For example: In the sequence: 4, 8, 12, 16, 20, 24; what is it’s arithmetic means. Solution: a. The arithmetic mean is a term between the first term and the last term. b. Thus, 8, 12, 16, and 20 are the arithmetic means of the sequence because these terms are between 4 and 24, which are the first and last term, respectively. 14 What is It The first and last terms of a finite arithmetic sequences are called arithmetic extremes, and the terms in between are called arithmetic means. In the sequence 4, 8, 12, 16, 20, 24; the terms 4 and 24 are the arithmetic extremes, while 8, 12, 16, and 20 are the arithmetic means. Also, 8 is the arithmetic mean of the arithmetic extremes, 4 and 12. The arithmetic mean between two numbers is sometimes called the average of two numbers. If more than one arithmetic means will be inserted between two arithmetic extremes, the formula for d, an −ak d = n−k , can be used. The formula for, d can be used to find the arithmetic means if more than one arithmetic means will be inserted between two arithmetic extremes. an − ak d= n−k Let’s Try! A. What is the arithmetic mean between 10 and 24? Solution a. Using the average formula, get the arithmetic mean of 10 and 24. 10+24 b. Thus, = 17 is the arithmetic mean. 2 Activity 1: Using the example above, solve for the arithmetic mean of each of the pairs of arithmetic extremes. Write your answer on your answer sheet. 1. 86, _____, 45 2. 135, _____, 170 3. 50, _____, - 30 4. 125, _____, 60 5. 43, _____, 89 15 B. Insert three arithmetic means between 8 and 16. Solution: a. If three arithmetic means will be inserted between 8 and 16, then a1= 8 and a5 = 16. 8, _____, _____, _____, 16 a 1 a2 a3 a4 a5 b. Using the formula for d, compute for the common difference. an − ak d= n−k a5 − a1 = 5−1 16 − 8 = 5−1 8 = 4 =2 c. The arithmetic means are a2, a3, and a4. a2 = a1 + d =8+2 = 10 a3 = a2 + d = 10 + 2 = 12 a4 = a3 + d = 12 + 2 = 14 d. Thus, the three arithmetic means between the arithmetic extremes, 8 and 16, are 10, 12, and 14. C. Insert two arithmetic means between √2 and 4√2 Solution: a. If two arithmetic means will be inserted between √2 and 4√2, then a1= √2 and a4 = 4√2. √2, _____, _____, 4√2 a 1 a2 a3 a4 b. Using the formula for d, compute for the common difference. an − ak d= n−k a4 − a1 = 4−1 4√2 − √2 = 4−1 3√2 = 3 =2 16 c. The arithmetic means are a2 and a3 a2 = a1 + d = √2 + √2 = 2√2 a3 = a2 + d = 2√2 + √2 = 3√2 d. Thus, the two arithmetic means between √2 and 4√2 are 2√2 and 3√2. D. Find the missing terms of the arithmetic sequence: _____, 6, _____, _____, 30. Solution: a. The arrangement of the terms tells that a 2 = 6 and a5 = 30. We are supposed to find for a1, a3, and a4. b. To find for the unknown, determine the common difference (d). an − ak d= n−k a5 − a2 = 5−2 30 − 6 = 5−2 24 = 3 =8 c. Thus, the value of a2, a3, and a4 are: a 1 = a2 – d =6–8 = –2 a 3 = a2 + d =6+8 = 14 a 4 = a3 + d = 14 + 8 = 22 Activity 2: Find the missing terms of the following sequence. Write answer on your answer sheet. 1. 15, _____, _____, _____, _____, 45 2. _____, 7, 13, _____, _____ 3. _____, 4, _____, 18, _____ 4. _____, 9, _____, _____, 36 5. 16, _____, _____, _____, 32 17 What’s More Let’s Do It! Write answer on your answer sheet. A. What is the arithmetic mean between the two given arithmetic extremes? 1. 5 and 19 2. 3𝑥 2 + 8 and 𝑥 2 – 6 3. -2 and 58 4. 2x + 3y and x – 5y 5. 13.8 and 15.6 B. Insert the specified number of arithmetic means between the two given arithmetic extremes. 1. Three arithmetic means between 2 and 22. 2. Four arithmetic means between 8 and 23. 3. Two arithmetic means between 41 and 95. 4. Two arithmetic means between -5 and 1. 5. Two arithmetic means between 97 and 172. What I Have Learned Answer the following questions on your answer sheet. 1. How do we find the arithmetic mean of two arithmetic extremes? 2. When two or more arithmetic means are inserted between two arithmetic extremes, how are they computed? 3. Do infinite sequences have arithmetic means? Why? What I Can Do A. What is the arithmetic mean between the given arithmetic extremes? 1. 19 and 7 3 7 2. and 11 11 3. 15x and 23x 4. 9√3 and 11√3 5. 6 - 7√7 and 2 + 3√7 18 B. Insert the specified number of arithmetic means between the given arithmetic extremes. 1. Three arithmetic means between 18 and 92. 2. Three arithmetic means between -14 and 6. 3. Four arithmetic means between 24 and -8. 4. Five arithmetic means between 6 and -18. 5. Two arithmetic means between 2√5 and 14√5. Assessment Choose the letter of your answer from the given options. Write your answer on your answer sheet. 1. Which term of the arithmetic sequence 5, 9, 13, 17, … is 409? a. 99th term b. 100th term c. 111th term d. 102th term 2. Find the nth term of the arithmetic sequence given the following given: 𝑎1 =5 d= 5 n=25 a. 25 term is115 th c. 25th term is120 b. 25th term is 125 d. 25th term is130 3. Which term of the arithmetic sequence 5, 9, 13, 17,….. is 401? a. 99th term b. 100th term c. 111th term d. 112th term 4. If three arithmetic means are inserted between -15 and 9, find the first of these arithmetic means. a. 3 b. -3 c. -6 d. -9 5. Find the 20th term of the arithmetic sequence 5, 9, 13, 17, 21,… a. 81 b. 80 c. 82 d. 87 6. If three arithmetic means are inserted between 11 and 39, find the second arithmetic mean. a. 18 b. 25 c. 32 d. 46 7. Which term of the arithmetic sequence 4, 1, -2, -5, … is -29? a. 9th term b. 10th term c. 11th term d. 12th term 8. What is the arithmetic mean between 15 and 40? a. 28.5 b. 29 c. 26 d. 27.5 19 9. What is the 8th term of the following arithmetic sequence: -5, -1, 3, 7, 11,…? a. 23 b. 19 c. 27 d. 22 10. Which of the following is the arithmetic mean between 2-√3 and 4 - √3? a. 3- √3 b. 3- 2√3 c. 3+ √3 b. 3+ 2√3 11. If a1 = -3 and a5 = 5. Find a10 ? a. 14 b. 15 c. 16 d. 17 12. If a3 = 11 and a5 = 7 and ak is the kth term of the sequence and ak = -9, then what is the value of k? a. 11 b. 12 c. 13 d. 14 13. Insert 3 arithmetic means between 8 and 16. a. 10, 12, 14 b. 9, 10, 11 c. 9, 11, 13 d. 12, 15, 16. 14. After one second, a rocket is 40 ft above the ground. After another second, it is 95 feet above the ground. Then after another second, it is already 150 feet above the ground. If it continues to rise at this rate, how many feet above the ground will the rocket be after 16 seconds? a. 780 ft b. 830 ft c. 855 ft d. 865 ft 15. Jose is the track and field representative of the Municipal NHS for the provincial meet. He begins training by running 5 miles during the first week, 6.5 miles during the second week, and 8 miles on the third week. If his training pattern continues, how far will he run on the tenth week? a. 18.5 miles b. 20 miles c. 21.5 miles d. 23 miles Additional Activity Solve the following word problems correctly on your answer sheet. 1. You have accepted a job with a salary of P27,000 a month during the first year. At the end of each year, you receive a P1500 raise. What is your monthly salary during the first six years? 2. An object is dropped from a plane and falls 32 feet during the first second. For each succeeding second, it falls 40 feet more than the distance covered in the preceding second. How far has it fallen after 11 seconds? 20 21 Lesson 1:What I Have Learned 1.the general formula is 𝑎𝑛 =𝑎1 + (n-1)d 2. and 3. by manipulating the general formula or by substitution to the general formula. Lesson I: What’s More What I Know: A. 1. B 2. A 1. 26 3. B Lesson 1:What' I Can Do 4. C 2. 41 A. 1. 39 5. D 13 3. - 6. B 2 2. 59 7. D 4. 53 8. D B. 1. 125.4 9. A 5. -130 2. 207 10. B 6. – 413 11. D 3. – 194 12. B 7. -11 13. C 4. 6.75 17 14. C 8. – or – 8.5 5. 84 2 15. D C. 1. 12 B. 1. 26 2. 103 2. 41 3. 72 3. -30 4. 14 4. 14 5. -5/4 or – 1.25 5. -400 Answer Key 22 Assessment: 1. D Lesson 2:What' I Can Do 2. B What is It 3. B A.1. 13 4. D Activity 1 5. A 2. 5/11 1. 65.5 or 131/2 6. B 2. 152.5 or 305/2 3. 19x 7. D 3. 10 8. D 4. 10√3 4. 92.5 or 185/2 9. A 5. 66 10. A 5. 4-2√7 11. B Activity 2 B. 1. 36.5, 55, 73.5 12. C 13. A 2. -9, -4, 1 1. 15, 21, 27, 33, 39, 45 14. D 2. 1, 7, 13, 19, 25 15. A 3. 17.6, 11.2, 4.8, 1.6 3. -3, 4, 11, 18, 25 4. 0, 9, 18, 27, 36 4. 2, -2, -6, -10, -14 5. 16, 20, 24, 28, 32 5. 6√5, 10√5 Additional Activity: Lesson 2: What I Have Lesson 2:What’s More: 1. 34,500.00- Learned. A. B. expected salary after six years. Depends on 1. 12 1. 7, 12, 17 2. After 11 seconds, students respond. 2. 2𝑥 2 + 1 2. 11, 14, 17, 20 the object is 3. 28 3. 59, 77 dropped 112 feet. 3 4. x- y 4. -3, -1 2 5. 14.7 5. 122, 147 References Callanta, Melvin M., 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. 23 For inquiries or feedback, please write or call: Department of Education – (Bureau/Office) (Office Address) Telefax: Email Address:

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