Grade 10 Mathematics Learner's Module PDF
Document Details
2015
Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, Rowena S. Perez, Concepcion S. Ternida
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This is a learner's module for Grade 10 mathematics, written by educators in the Philippines, from the Department of Education. The module covers various parts of mathematics, including sequences, and aims to help students reach specified standards.
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10 PY Mathematics O...
10 PY Mathematics O C Learner’s Module D Unit 1 E EP This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of D Education at [email protected]. We value your feedback and recommendations. Department of Education Republic of the Philippines All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Mathematics – Grade 10 Learner’s Module First Edition 2015 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. DepEd is represented by the Filipinas Copyright Licensing Society (FILCOLS), Inc. in seeking permission to use these materials from their respective copyright owners. All means have been exhausted in seeking permission to use these materials. The publisher and authors do not represent nor claim ownership over them. PY Only institution and companies which have entered an agreement with FILCOLS and only within the agreed framework may copy this Learner’s Module. Those who have not entered in an agreement with FILCOLS must, if they wish to copy, contact the publisher and authors directly. Authors and publishers may email or contact FILCOLS at [email protected] or (02) 439-2204, respectively. O Published by the Department of Education Secretary: Br. Armin A. Luistro FSC Undersecretary: Dina S. Ocampo, PhD C Development Team of the Learner’s Module Consultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, and D Rosemarievic Villena-Diaz, PhD Authors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, E Rowena S. Perez, and Concepcion S. Ternida Editor: Maxima J. Acelajado, PhD EP Reviewers: Maria Alva Q. Aberin, PhD, Maxima J. Acelajado, PhD, Carlene P. Arceo, PhD, Rene R. Belecina, PhD, Dolores P. Borja, Agnes D. Garciano, Phd, Ma. Corazon P. Loja, Roger T. Nocom, Rowena S. Requidan, and Jones A. Tudlong, PhD Illustrator: Cyrell T. Navarro D Layout Artists: Aro R. Rara and Ronwaldo Victor Ma. A. Pagulayan Management and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr., Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr. Printed in the Philippines by REX Book Store Department of Education-Instructional Materials Council Secretariat (DepEd-IMCS) Office Address: 5th Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600 Telefax: (02) 634-1054, 634-1072 E-mail Address: [email protected] All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Introduction This material is written in support of the K to 12 Basic Education Program to ensure attainment of standards expected of students. In the design of this Grade 10 materials, it underwent different processes - development by writers composed of classroom teachers, school heads, supervisors, specialists from the Department and other institutions; validation by experts, academicians, and practitioners; revision; content review and language editing by members of Quality Circle Reviewers; and PY finalization with the guidance of the consultants. There are eight (8) modules in this material. Module 1 – Sequences O Module 2 – Polynomials and Polynomial Equations Module 3 – Polynomial Functions Module 4 – Circles C Module 5 – Plane Coordinate Geometry Module 6 – Permutations and Combinations Module 7 – Probability of Compound Events Module 8 – Measures of Position D With the different activities provided in every module, may you find this E material engaging and challenging as it develops your critical-thinking and problem-solving skills. EP D All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Table of Contents Unit 1 Module 1: Sequences................................................................................... 1 Lessons and Coverage............................................................................ 2 Module Map............................................................................................. 3 Pre-Assessment...................................................................................... 4 Learning Goals and Targets.................................................................... 8 Lesson 1: Arithmetic Sequences..................................................................... 9 Activity 1........................................................................................ 9 Activity 2...................................................................................... 11 Activity 3...................................................................................... 11 Activity 4...................................................................................... 12 PY Activity 5...................................................................................... 13 Activity 6...................................................................................... 14 Activity 7...................................................................................... 15 Activity 8...................................................................................... 16 Activity 9...................................................................................... 18 O Activity 10.................................................................................... 18 Activity 11.................................................................................... 19 Activity 12.................................................................................... 20 C Activity 13.................................................................................... 21 Activity 14.................................................................................... 23 Summary/Synthesis/Generalization............................................................. 25 Lesson 2: Geometric and Other Sequences................................................. 26 D Activity 1...................................................................................... 26 Activity 2...................................................................................... 27 Activity 3...................................................................................... 28 E Activity 4...................................................................................... 28 Activity 5...................................................................................... 29 EP Activity 6...................................................................................... 31 Activity 7...................................................................................... 37 Activity 8...................................................................................... 39 Activity 9...................................................................................... 40 Activity 10.................................................................................... 41 Activity 11.................................................................................... 42 D Activity 12.................................................................................... 43 Activity 13.................................................................................... 44 Summary/Synthesis/Generalization............................................................. 46 Glossary of Terms........................................................................................ 47 References and Website Links Used in this Module................................... 48 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Module 2: Polynomials and Polynomial Equations............................. 49 Lessons and Coverage.......................................................................... 50 Module Map........................................................................................... 50 Pre-Assessment..................................................................................... 51 Learning Goals and Targets................................................................... 56 Lesson 1: Division of Polynomials................................................................ 57 Activity 1...................................................................................... 57 Activity 2...................................................................................... 58 Activity 3...................................................................................... 60 Activity 4...................................................................................... 63 Activity 5...................................................................................... 64 Activity 6...................................................................................... 65 Activity 7...................................................................................... 65 PY Activity 8...................................................................................... 66 Activity 9...................................................................................... 67 Activity 10.................................................................................... 68 Summary/Synthesis/Generalization............................................................. 69 Lesson 2: The Remainder Theorem and Factor Theorem.......................... 70 O Activity 1...................................................................................... 70 Activity 2...................................................................................... 71 Activity 3...................................................................................... 72 C Activity 4...................................................................................... 74 Activity 5...................................................................................... 76 Activity 6...................................................................................... 76 Activity 7...................................................................................... 77 D Activity 8...................................................................................... 78 Activity 9...................................................................................... 79 Activity 10.................................................................................... 80 E Summary/Synthesis/Generalization............................................................. 81 Lesson 3: Polynomial Equations................................................................... 82 EP Activity 1...................................................................................... 82 Activity 2...................................................................................... 83 Activity 3...................................................................................... 84 Activity 4...................................................................................... 85 Activity 5...................................................................................... 87 D Activity 6...................................................................................... 88 Activity 7...................................................................................... 89 Activity 8...................................................................................... 91 Activity 9...................................................................................... 91 Activity 10.................................................................................... 92 Activity 11.................................................................................... 92 Activity 12.................................................................................... 93 Activity 13.................................................................................... 93 Activity 14.................................................................................... 94 Activity 15.................................................................................... 95 Summary/Synthesis/Generalization............................................................. 96 Glossary of Terms......................................................................................... 96 List of Theorems Used in this Module......................................................... 96 References and Website Links Used in this Module................................... 97 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. I. INTRODUCTION PY O C E D “Kilos Kabataan” EP In her first public address, the principal mentioned about the success of the recent “Brigada Eskwela.” Because of this success, the principal challenged the students, especially the Grade 9 and Grade 10 students, to extend the same service to their community by having a one- Saturday community clean-up which the principal called “Kilos Kabataan D Project.” Volunteers have to sign up until 5 p.m. for the project. Accepting the principal’s challenge, 10 students immediately signed up for the clean- up. After 10 minutes, there were already 15 who had signed up. After 10 more minutes, there were 20, then 25, 30, and so on. Amazed by the students’ response to the challenge, the principal became confident that the youth could be mobilized to create positive change. The above scenario illustrates a sequence. In this learning module, you will know more about sequences, and how the concept of a sequence is utilized in our daily lives. 1 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. II. LESSONS AND COVERAGE In this module, you will learn more about sequences when you take the following lessons: Lesson 1 – Arithmetic Sequences Lesson 2 – Geometric and Other Sequences In these lessons you will learn to: generate and describe patterns PY find the next few terms of a sequence find the general or nth term of a sequence illustrate an arithmetic sequence determine the nth term of a given arithmetic sequence Lesson 1 find the arithmetic means between terms of an O arithmetic sequence determine the sum of the first n terms of a given arithmetic sequence C solve problems involving arithmetic sequence illustrate a geometric sequence D differentiate a geometric sequence from an arithmetic sequence determine the nth term of a given geometric sequence E find the geometric means between terms of a geometric sequence EP Lesson 2 determine the sum of the first n terms of a geometric sequence determine the sum of the first n terms of an infinite geometric sequence D illustrate other types of sequences like harmonic sequence and Fibonacci sequence solve problems involving geometric sequence 2 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Sequences Arithmetic Geometric Other Types of Sequences Sequences Sequences PY O Finding the Next Term C Finding the nth Term D Finding the Arithmetic/Geometric Means E EP Finding the Sum of the First n Terms D Solving Real-Life Problems 3 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. III. PRE-ASSESSMENT Part 1 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. 1. What is the next term in the geometric sequence 4, 12, 36? A. 42 B. 54 C. 72 D. 108 PY 13 7 15 2. Find the common difference in the arithmetic sequence 3, , , ,... 4 2 4 1 3 5 A. B. C. D. 4 O 4 4 2 3. Which set of numbers is an example of a harmonic sequence? 1 1 1 1 C 1 1 1 1 A. , , , , C. , , , 2 2 2 2 3 9 27 81 D 1 2 2 2 B. , 1, 2, 4 D. 2, , , 2 3 5 7 E 4. What is the sum of all the odd integers between 8 and 26? EP A. 153 B. 151 C. 149 D. 148 5. If three arithmetic means are inserted between 11 and 39, find the second arithmetic mean. A. 18 B. 25 C. 32 D. 46 D 6. If three geometric means are inserted between 1 and 256, find the third geometric mean. A. 64 B. 32 C. 16 D. 4 1 1 1 1 7. What is the next term in the harmonic sequence , , , ,...? 11 15 19 23 1 1 A. 27 B. 25 C. D. 25 27 4 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. 8. Which term of the arithmetic sequence 4, 1, 2, 5 ,... is 29 ? A. 9th term B. 10th term C. 11th term D. 12th term 2 2 9. What is the 6th term of the geometric sequence , , 2, 10,...? 25 5 A. 25 B. 250 C. 1250 D. 2500 10. The first term of an arithmetic sequence is 2 while the 18th term is 87. Find the common difference of the sequence. A. 7 B. 6 C. 5 D. 3 PY 11. What is the next term in the Fibonacci sequence 1, 1, 2, 3, 5, 8,...? A. 13 B. 16 C. 19 D. 20 O 12. Find the sum of the geometric sequence where the first term is 3, the last term is 46 875, and the common ratio is 5. C A. 58 593 B. 58 594 C. 58 595 D. 58 596 13. Find the eighth term of a geometric sequence where the third term is 27 and the common ratio is 3. D A. 2187 B. 6561 C. 19 683 D. 59 049 E 14. Which of the following is the sum of all the multiples of 3 from 15 to 48? EP A. 315 B. 360 C. 378 D. 396 n2 1 15. What is the 7th term of the sequence whose nth term is an ? n2 1 D 24 23 47 49 A. B. C. D. 25 25 50 50 16. What is the nth term of the arithmetic sequence 7, 9, 11, 13, 15, 17,..? A. 3n 4 B. 4n 3 C. n 2 D. 2n 5 1 1 1 1 17. What is the nth term of the harmonic sequence , , , ,...? 2 4 6 8 1 1 1 1 A. B. 2 C. D. n 1 n 1 2n 4n 2 5 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. 18. Find p so that the numbers 7p 2, 5p 12, 2p 1,... form an arithmetic sequence. A. 8 B. 5 C. 13 D. 23 3 9 27 81 19. What is the sum of the infinite geometric series ...? 4 16 64 256 3 3 A. 3 B. 1 C. D. 4 7 20. Find k so that the numbers 2k 1, 3k 4, and 7k 6 form a geometric sequence. PY A. 2; -1 B. -2; 1 C. 2; 1 D. -2; -1 21. Glenn bought a car for Php600,000. The yearly depreciation of his car is 10% of its value at the start of the year. What is its value after 4 years? O A. Php437,400 B. Php438,000 C. Php393,660 D. Php378,000 22. During a free-fall, a skydiver jumps 16 feet, 48 feet, and 80 feet on the C first, second, and third fall, respectively. If he continues to jump at this rate, how many feet will he have jumped during the tenth fall? A. 304 B. 336 C. 314 928 D. 944 784 D 23. Twelve days before Valentine’s Day, Carl decided to give Nicole flowers according to the Fibonacci sequence. On the first day, he sent E one red rose, on the second day, two red roses, and so on. How many roses did Nicole receive during the tenth day? EP A. 10 B. 55 C. 89 D. 144 24. A new square is formed by joining the midpoints of the consecutive sides of a square 8 inches on a side. If the process is continued until there are already six squares, find the sum of the areas of all squares D in square inches. A. 96 B. 112 C. 124 D. 126 25. In President Sergio Osmeña High School, suspension of classes is announced through text brigade. One stormy day, the principal announces the suspension of classes to two teachers, each of whom sends this message to two other teachers, and so on. Suppose that text messages were sent in five rounds, counting the principal’s text message as the first, how many text messages were sent in all? A. 31 B. 32 C. 63 D. 64 6 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Part II Read and understand the situation below, then answer the questions or perform the tasks that follow. Hold on to HOPE Because of the super typhoon Yolanda, there was a big need for blood donors, medicines, doctors, nurses, medical aides, or any form of medical assistance. The Red Cross planned to involve different agencies, organizations, and offices, public and private, local and international, in their project to have massive medical services. The Red Cross contacted first PY three of the biggest networks, and each of these networks contacted three other networks, and agencies, organizations, and offices, and so on, until enough of these were contacted. It took one hour for an organization to contact three other organizations and all the contacts made were completed O within 4 hours. Assume that no group was contacted twice. 1. Suppose you are one of the people in the Red Cross who visualized this project. How many organizations do you think were contacted in C the last round? How many organizations were contacted within 4 hours? D 2. Make a table to represent the number of organizations, agencies, and offices who could have been contacted in each round. E 3. Write an equation to represent the situation. Let the independent EP variable be the number of rounds and the dependent variable be the number of organizations, agencies, and offices that were contacted in that round. D 4. If another hour was used to contact more organizations, how many additional organizations, agencies, and offices could be contacted? 5. Use the given information in the above situation to formulate problems involving these concepts. 6. Write the necessary equations that describe the situations or problems that you formulated. 7. Solve the problems that you formulated. 7 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Rubric for the Equations Formulated and Solved Score Descriptors 4 Equations are properly formulated and solved correctly. Equations are properly formulated but not all are solved 3 correctly. Equations are properly formulated but are not solved 2 correctly. 1 Equations are properly formulated but are not solved at all. Rubric for the Problems Formulated and Solved PY Score Descriptors Poses a more complex problem with two or more solutions and communicates ideas unmistakably, shows in-depth 6 comprehension of the pertinent concepts and/or processes O and provides explanation wherever appropriate Poses a more complex problem and finishes all significant 5 C parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes D Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows 4 in-depth comprehension of the pertinent concepts and/or E processes Poses a complex problem and finishes most significant parts EP of the solution and communicates ideas unmistakably, shows 3 comprehension of major concepts although neglects or misinterprets less significant ideas or details Poses a problem and finishes some significant parts of the 2 solution and communicates ideas unmistakably but shows D gaps on the theoretical comprehension Poses a problem but demonstrates minor comprehension, 1 not being able to develop an approach Source: D.O. #73, s. 2012 IV. LEARNING GOALS AND TARGETS After using this module, you should be able to demonstrate understanding of sequences like arithmetic sequences, geometric sequences, and other types of sequences and solve problems involving sequences. 8 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. In this lesson, you will work with patterns. Recognizing and PY extending patterns are important skills needed for learning concepts related to an arithmetic sequence. Activity 1: O Each item below shows a pattern. Answer the given questions. C 1. What is the next shape? D , , , , , , , , , , , , , ___ E 2. What is the next number? EP What is the 10th number? 0, 4, 8, 12, 16, ____ D 3. What is the next number? What is the 8th number? 9, 4, -1, -6, -11, ____ 9 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. 4. What is the next number? What is the 12th number? 1, 3, 9, 27, 81, _____ 5. What is the next number? What is the 7th number? 160, 80, 40, 20, 10, _____ The set of shapes and the sets of numbers in the above activity are PY called sequences. Were you able to find patterns and get the next number in the sequence? Let us now give the formal definition of a sequence. O What is a sequence? C A sequence is a function whose domain is the finite set {1, 2, 3,…, n} or the infinite set {1, 2, 3,… }. D n 1 2 3 4 5 Example: E a n 3 1 1.5 10 EP This finite sequence has 5 terms. We may use the notation a1, a2 , a3 ,..., an to denote a 1 , a 2, a 3 ,..., a n , respectively. In Grade 10, we often encounter sequences that form a pattern such D as that found in the sequence below. n 1 2 3 4... Example: an 4 7 10 13... The above sequence is an infinite sequence where an 3n 1 10 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. In the next two activities, you will learn more about sequences. A general term or nth term will be given to you and you will be asked to give the next few terms. You will also be asked to give the nth term or the rule for a particular sequence. You may now start with Activity 2. Activity 2: Find the first 5 terms of the sequence given the nth term. 1. an n 4 PY 2. a n 2n 1 3. a n 12 3n 4. an 3n O 2 n 5. an C How did you find the activity? Did you find it easy to give the first 5 terms of each sequence? In Activity 3, you will be given the terms of a sequence and you will be asked to find its nth term. You may now do D Activity 3. E Activity 3: EP What is the nth term for each sequence below? 1. 3, 4, 5, 6, 7,... 2. 3, 5, 7, 9, 11,... D 3. 2, 4, 8, 16, 32,... 4. -1, 1, -1, 1, -1,... 1 1 1 1 5. 1, , , , ,... 2 3 4 5 In the activities you have just done, you were able to enumerate the terms of a sequence given its nth term and vice versa. Knowing all these will enable you to easily understand a particular sequence. This sequence will be discussed after doing the following activity. 11 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Activity 4: We need matchsticks for this group activity. Form a group of 3 students. 1. Below are squares formed by matchsticks. 2. Count the number of matchsticks in each figure and record the results in a table. PY number of squares 1 2 3 4 5 6 7 8 9 10 number of matchsticks O C 1. Is there a pattern in the number of matchsticks? If there is, describe it. 2. How is each term (number of matchsticks) found? D 3. What is the difference between any two consecutive terms? E How was the activity? What new thing did you learn from the activity? EP The above activity illustrates a sequence where the difference between any two consecutive terms is a constant. This constant is called the common difference and the said sequence is called an arithmetic sequence. D An arithmetic sequence is a sequence where every term after the first is obtained by adding a constant called the common difference. The sequences 1, 4, 7, 10,... and 15, 11, 7, 3,... are examples of arithmetic sequences since each one has a common difference of 3 and 4, respectively. Is the meaning of arithmetic sequence clear to you? Are you ready to learn more about arithmetic sequences? If so, then you have to perform the next activity. 12 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Activity 5: Let us go back to Activity 4. With your groupmates, take a look at the completed table below. number of squares 1 2 3 4 5 6 7 8 9 10 number of matchsticks 4 7 10 13 16 19 22 25 28 31 Let us take the number of matchsticks 4, 7, 10, 13, 16, 19, 22, 25, 28, and 31. We see that the number of matchsticks forms an arithmetic PY sequence. Suppose we want to find the 20th, 50th, and 100th terms of the sequence. How do we get them? Do you think a formula would help? If so, we could find a formula for the nth term of the sequence. In this case, it will not be difficult since we know the common difference of the sequence. O Let us take the first four terms. Let a1 4, a2 7, a3 10, a4 13. How do we obtain the second, third, and fourth terms? C Consider the table below and complete it. Observe how each term is rewritten. D a1 a2 a3 a4 a5 a6 a7 a8... an E 4 4+3 4+3+3 4+3+3+3... EP How else can we write the terms? Study the next table and complete it. a1 a2 a3 a4 a5 a6 a7 a8... an D 4+0(3) 4+1(3) 4+2(3) 4+3(3)... What is a5 ? a20 ? a50 ? What is the formula for determining the number of matchsticks needed to form n squares? In general, the first n terms of an arithmetic sequence with a1 as first term and d as common difference are a1, a1 d, a1 2d,..., a1 n 1 d. 13 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. If a1 and d are known, it is easy to find any term in an arithmetic sequence by using the rule an a1 n 1 d. Example: What is the 10th term of the arithmetic sequence 5, 12, 19, 26,...? Solution: Since a1 5 and d 7, then a10 5 10 1 7 68. How did you find the activity? The rule for finding the nth term of an arithmetic sequence is very useful in solving problems involving arithmetic sequence. PY Activity 6: A. Find the missing terms in each arithmetic sequence. 1. 3, 12, 21, __, __, __ O 2. 8, 3, 2 , __, __ C 3. 5, 12, __, 26, __ 4. 2, __, 20, 29, __ D 5. __, 4, 10, 16, __ 6. 17, 14, __, __, 5 7. 4, __, __, 19, 24,... E 8. __, __, __, 8, 12, 16 9. 1, __, __, __, 31, 39 EP 10. 13, __, __, __, 11, 17 B. Find three terms between 2 and 34 of an arithmetic sequence. D Were you able to get the missing terms in each sequence in Part A? Were you able to get the 3 terms in Part B? Let us discuss a systematic way of finding missing terms of an arithmetic sequence. Finding a certain number of terms between two given terms of an arithmetic sequence is a common task in studying arithmetic sequences. The terms between any two nonconsecutive terms of an arithmetic sequence are known as arithmetic means. 14 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Example: Insert 4 arithmetic means between 5 and 25. Solution: Since we are required to insert 4 terms, then there will be 6 terms in all. Let a1 5 and a6 25. We will insert a2 , a3 , a4 , a5 as shown below: 5, a2 , a3 , a4 , a5 , 25 We need to get the common difference. Let us use a6 a1 5d to solve for d. Substituting the given values for a6 and a1 , we obtain 25 5 5d. PY So, d 4. Using the value of d, we can now get the values of a2 , a3 , a4 , and a5. 5 4 1 Thus, a2 5 4 2 9, a3 5 4 3 13, a4 17, and 5 4 4 a5 21. O The 4 arithmetic means between 5 and 25 are 9, 13, 17, and 21. C At this point, you know already some essential things about arithmetic sequence. Now, we will learn how to find the sum of the first n terms of an arithmetic sequence. Do Activity 7. E D Activity 7: EP What is the sum of the terms of each finite sequence below? 1. 1, 4, 7, 10 2. 3, 5, 7, 9, 11 D 3. 10, 5, 0, -5, -10, -15 4. 81, 64, 47, 30, 13, -4 5. -2, -5, -8, -11, -14, -17 15 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Activity 8: What is 1 + 2 + 3 +... + 50 + 51 +... + 98 + 99 + 100? A famous story tells that this was the problem given by an elementary school teacher to a famous mathematician to keep him busy. Do you know that he was able to get the sum within seconds only? Can you beat that? His name was Karl Friedrich Gauss (1777-1885). Do you know how he did it? Let us find out by doing the activity below. Think-Pair-Share PY Determine the answer to the above problem. Then look for a partner and compare your answer with his/her answer. Discuss with him/her your technique (if any) in getting the answer quickly. Then with your partner, O answer the questions below and see if this is similar to your technique. 1. What is the sum of each of the pairs 1 and 100, 2 and 99, 3 and 98,..., 50 and 51? C 2. How many pairs are there in #1? 3. From your answers in #1 and #2, how do you get the sum of the D integers from 1 to 100? 4. What is the sum of the integers from 1 to 100? E Let us now denote the sum of the first n terms of an arithmetic sequence a1 a2 a3 ... an by Sn. EP We can rewrite the sum in reverse order, that is, Sn an an 1 an 2 ... a1. Rewriting the two equations above using their preceding terms and the difference d, we would have D Equation 1: Sn a1 a1 d a1 2d ... a1 n 1 d Equation 2 : Sn an an d an 2d ... an n 1 d Adding equation 1 and equation 2, we get 2Sn a1 an a1 an a1 an ... a1 an . Sn n a1 an . Since there are n terms of the form a1 an , then 2 16 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. n Dividing both sides by 2, we have Sn a1 an . 2 Now, since we also know that an a1 n 1 d, then by substitution, we have n a1 a1 n 1 d n S n 2 2 2a1 (n 1)d . or S n Example 1: Find the sum of the first 10 terms of the arithmetic sequence 5, 9, 13, 17,... 10 Solution: S10 2 5 10 1 4 230 2 PY Example 2: Find the sum of the first 20 terms of the arithmetic sequence 2, 5, 8, 11,... 20 Solution: S20 2 2 20 1 3 610 O 2 C How did you find Activity 7? Did you learn many things about arithmetic sequences? D http://coolmath.com/algebra/19-sequences- E series/05-arithmetic-sequences-01.html Learn more about arithmetic http://www.mathisfun.com/algebra/sequences- EP sequences through the web. series.html You may open the following http://www.mathguide.com/lessons/SequenceArit hmetic.html#identify links: D 17 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Your goal in this section is to apply the key concepts of arithmetic sequence. Use the mathematical ideas and the examples presented in the preceding section to answer the activities provided. Activity 9: Which of the following sequences is an arithmetic sequence? Why? PY 1.3, 7, 11, 15, 19 2.4, 16, 64, 256 3.48, 24, 12, 6, 3,... 4.1, 4, 9, 16, 25, 36 O 1 1 5. 1, , 0, 2 2 6. 2, 4, 8, 16,... 7. 1, 0, 1, 2, , 3 C 1 1 1 1 8. , , , ,... 2 3 4 5 D x x 9. 3 x, x, , ,... 3 9 E 10. 9.5, 7.5, 5.5, 3.5,... EP Did you find it easy to determine whether a sequence is arithmetic or not? Were you able to give a reason why? The next activity will assess your skill in using the nth term of an arithmetic sequence. You may start the activity now. D Activity 10: Use the nth term of an arithmetic sequence an a1 n 1 d to answer the following questions. 1. Find the 25th term of the arithmetic sequence 3, 7, 11, 15, 19,... 2. The second term of an arithmetic sequence is 24 and the fifth term is 3. Find the first term and the common difference. 18 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. 3. Give the arithmetic sequence of 5 terms if the first term is 8 and the last term is 100. 1 4. Find the 9th term of the arithmetic sequence with a1 10 and d . 2 5. Find a1 if a8 54 and a9 60. 6. How many terms are there in an arithmetic sequence with a common difference of 4 and with first and last terms 3 and 59, respectively? 7. Which term of the arithmetic sequence is 18, given that a1 7 and a2 2? 8. How many terms are in an arithmetic sequence whose first term is -3, PY common difference is 2, and last term is 23? 9. What must be the value of k so that 5k 3, k 2, and 3k 11 will form an arithmetic sequence? 10. Find the common difference of the arithmetic sequence with a4 10 O and a11 45. Did you find the activity challenging? The next activity is about C finding arithmetic means. Remember the nth term of an arithmetic sequence. You may now do Activity 11. D Activity 11: E A. Insert the indicated number of arithmetic means between the given first EP and last terms of an arithmetic sequence. 1. 2 and 32 2. 6 and 54 3. 68 and 3 D 4. 10 and 40 1 5. and 2 2 6. –4 and 8 7. –16 and –8 1 11 8. and 3 3 9. a and b 10. x y and 4x 2y 19 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. B. Solve the following problems. 1. The arithmetic mean between two terms in an arithmetic sequence is 39. If one of these terms is 32, find the other term. 2. If five arithmetic means are inserted between 9 and 9, what is the third mean? 3. What are the first and last terms of an arithmetic sequence when its arithmetic means are 35, 15, and 5? 4. Find the value of x if the arithmetic mean of 3 and 3 x 5 is 8. 5. Find the value of a when the arithmetic mean of a 7 and a 3 is 3a 9. PY Did you find the nth term of an arithmetic sequence helpful in finding the arithmetic means? O The next activity is about finding the sum of the terms of an arithmetic sequence. You may now proceed. C Activity 12: A. Find the sum of each of the following. D 1. integers from 1 to 50 E 2. odd integers from 1 to 100 3. even integers between 1 and 101 EP 4. first 25 terms of the arithmetic sequence 4, 9, 14, 19, 24,... 5. multiples of 3 from 15 to 45 6. numbers between 1 and 81 which are divisible by 4 7. first 20 terms of the arithmetic sequence –16, –20, –24, … D 8. first 10 terms of the arithmetic sequence 10.2, 12.7, 15.2, 17.7, … 9. 1 + 5 + 9 + … + 49 + 53 1 3 5 17 19 10. ... 2 2 2 2 2 B. The sum of the first 10 terms of an arithmetic sequence is 530. What is the first term if the last term is 80? What is the common difference? C. The third term of an arithmetic sequence is –12 and the seventh term is 8. What is the sum of the first 10 terms? 20 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. D. Find the sum of the first 25 multiples of 8. E. Find the sum of the first 12 terms of the arithmetic sequence whose general term is a n 3n 5. Were you able to answer Activity 12? In this section, you were provided with activities to assess your knowledge and skill in what you learned in the previous section. Now that you know the important ideas about arithmetic sequences, let us go deeper by moving to the next section. PY O Activity 13: C D Do each of the following. 1. Mathematically speaking, the next term cannot be determined by E giving only the first finite number of terms of a general sequence. Explain this fact by giving an example. EP 2. Make a concept map for arithmetic sequences. 3. Using the formula for arithmetic sequence, an a1 n 1 d, D give problems where the unknown value is (a) a1 , (b) an , (c) d and show how each can be found. 4. What should be the value of x so that x + 2, 3x – 2, 7x – 12 will form an arithmetic sequence? Justify your answer. 5. Find the value of x when the arithmetic mean of x + 2 and 4x + 5 is 3x + 2. 21 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. 6. It is alarming that many people now are being infected by HIV. As the president of the student body in your school, you invited people to give a five-day series of talks on HIV and its prevention every first Friday of the month from 12 noon to 1 p.m. in the auditorium. On the first day, 20 students came. Finding the talk interesting, these 20 students shared the talk to other students and 10 more students came on the second day, another 10 more students came 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. PY b. Represent the data in the table using a formula. Use the formula to justify your data in the table. O 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 C many students in all attended the talk on HIV? Were you able to accomplish the activity? How did you find it? D You may further assess your knowledge and skill by trying another activity. E EP Try This: After a knee surgery, your trainer tells you to return to your jogging program slowly. He suggests jogging for 12 minutes each day for the first D week. Each week, thereafter, he suggests that you increase that time by 6 minutes per day. On what week will it be before you are up to jogging 60 minutes per day? Were you able to solve the problem? Now that you have a deeper understanding of the topic, you are now ready to do the tasks in the next section. 22 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Your goal in this section is to apply what you learned to real-life situations. You will be given a task which will demonstrate your understanding of arithmetic sequences. Activity 14: PY In groups of five, create a well-developed Reality Series considering the following steps: 1. Choose a real-life situation which involves arithmetic sequences. You O could research online or create your own. Be sure to choose what interests your group the most to make your Reality Series not only interesting but also entertaining. C 2. Produce diagrams or pictures that will help others see what is taking place in the situation or the scenario that you have chosen. 3. Prepare the necessary table to present the important data in your D situation and the correct formula and steps to solve the problem. 4. Show what you know about the topic by using concepts about arithmetic sequences to describe the situation. For example, show how E to find the nth term of your arithmetic sequence or find the sum of the first n terms. Write your own questions about the situation and be EP ready with the corresponding answers. 5. Present your own Reality Series in the class. How did the task help you realize the importance of the topic in real life? D 23 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Rubric for the Written Report about Chosen Real-Life Situation Score Descriptors The written report is completely accurate and logically presented/designed. It includes facts, concepts, and 5 computations involving arithmetic sequences. The chosen real- life situation is very timely and interesting. The written report is generally accurate and the presentation/design reflects understanding of arithmetic 4 sequences. Minor inaccuracies do not affect the overall results. The chosen real-life situation is timely and interesting. PY The written report is generally accurate but the presentation/design lacks application of arithmetic sequences. 3 The chosen real-life situation is somehow timely and interesting. The written report contains major inaccuracies and significant O 2 errors in some parts. The chosen real-life situation is not timely and interesting. 1 There is no written report made. C Rubric for the Oral Presentation D Score Descriptors E Oral presentation is exceptionally clear, thorough, fully 5 supported with concepts and principles of arithmetic EP sequences, and easy to follow. Oral report is generally clear and reflective of students’ 4 personalized ideas, and some accounts are supported by mathematical principles and concepts of arithmetic sequences. Oral report is reflective of something learned; it lacks clarity and D 3 accounts have limited support. Oral report is unclear and impossible to follow, is superficial, 2 and more descriptive than analytical. 1 No oral report was presented. 24 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. SUMMARY/SYNTHESIS/GENERALIZATION This lesson is about arithmetic sequences and how they are illustrated in real life. You learned to: generate patterns; determine the nth term of a sequence; describe an arithmetic sequence, and find its nth term; determine the arithmetic means of an arithmetic sequence; find the sum of the first n terms of an arithmetic sequence; and solve real-life problems involving arithmetic sequence. PY O C E D EP D 25 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. The previous lesson focused on arithmetic sequences. In this lesson, you will also learn about geometric sequences and the process PY on how they are generated. You will also learn about other types of sequences. Activity 1: O Find the ratio of the second number to the first number. 1. 2, 8 C 2. –3, 9 1 3. 1, D 2 4. –5, –10 5. 12, 4 E 6. –49, 7 1 1 EP 7. , 4 2 8. a2, a3 9. k–1, k 10. 3m, 3mr D You need the concept of ratio in order to understand the next kind of sequence. We will explore that sequence in the next activity. Do the next activity now. 26 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Activity 2: Do the activity with a partner. One of you will perform the paper folding while the other will do the recording in the table. 1. Start with a big square from a piece of paper. Assume that the area of the square is 64 square units. 2. Fold the four corners to the center of the square and find the area of the resulting square. 3. Repeat the process three times and record the results in the table below. 3 PY Square 1 2 Area O C 1. What is the area of the square formed after the first fold? Second fold? Third fold? 2. Is there a pattern in the areas obtained after 3 folds? D 3. You have generated a sequence of areas. What are the first 3 terms of the sequence? 4. Is the sequence an arithmetic sequence? Why? E 5. Using the pattern in the areas, what would be the 6th term of the sequence? EP The sequence 32, 16, 8, 4, 2, 1 is called a geometric sequence. A geometric sequence is a sequence where each term after the first is obtained by multiplying the preceding term by a nonzero constant called the D common ratio. The common ratio, r, can be determined by dividing any term in the sequence by the term that precedes it. Thus, in the geometric sequence 1 16 1 32, 16, 8, 4, 2,... , the common ratio is since . 2 32 2 The next activity will test whether you can identify geometric sequences or not. 27 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Activity 3: State whether each of the following sequences is geometric or not. 1. 5, 20, 80, 320,... 2. 7 2, 5 2, 3 2, 2,... 3. 5, –10, 20, –40 4. 1, 0.6, 0.36, 0.216,... 10 10 10 10 5. , , , 3 6 9 15 6. 4, 0, 0, 0, 0… PY Activity 4: Form a group of 3 members and answer the guide questions using the table. Problem: What are the first 5 terms of a geometric sequence whose first term O is 2 and whose common ratio is 3? Other Ways to Write the Terms Term C In Factored Form In Exponential Form a1 2 2 2 x 30 D a2 6 2x3 2 x 31 a3 18 2x3x3 2 x 32 E a4 54 2x3x3x3 2 x 33 a5 162 2x3x3x3x3 2 x 34 EP an ? D 1. Look at the two ways of writing the terms. What does 2 represent? 2. For any two consecutive terms, what does 3 represent? 3. What is the relationship between the exponent of 3 and the position of the term? 4. If the position of the term is n, what must be the exponent of 3? 5. What is an for this sequence? 6. In general, if the first term of a geometric sequence is a1 and the common ratio is r, what is the nth term of the sequence? 28 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. What did you learn in the activity? Given the first term a1 and the common ratio r of a geometric sequence, the nth term of a geometric sequence is an a1r n -1. Example: What is the 10th term of the geometric sequence 8, 4, 2, 1,...? 9 1 1 1 1 Since r , then a10 8 8 . Solution: 2 2 512 64 In the next activity, you will find the nth term of a geometric sequence, a skill that is useful in solving other problems involving geometric sequences. Do the next activity. PY Activity 5: A. Find the missing terms in each geometric sequence. 1. 3, 12, 48, O __, __ C... 2. __, __, 32, 64, 128, __ D 3. 120, 60, 30, __, __, __ 4. 5, __, 20, 40, __, E __, __ 5. __, 4, 12, 36, EP 6. –2, __, __, –16 –32 –64... 7. 256, __, __, –32 16, D 1 8. 27, 9, __, __, 3 1 9. , __, __, __, 64, 256 4 10. 5x2 __, 5x6 5x8 __ ,... B. Insert 3 terms between 2 and 32 of a geometric sequence. 29 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Were you able to answer the activity? Which item in the activity did you find challenging? Let us now discuss how to find the geometric means between terms of a geometric sequence. Inserting a certain number of terms between two given terms of a geometric sequence is an interesting activity in studying geometric sequences. We call the terms between any two given terms of a geometric sequence the geometric means. Example: Insert 3 geometric means between 5 and 3125. PY Solution: Let a1 5 and a5 3125. We will insert a2 , a3 , and a4. O Since a5 a1r 4 , then 3125 5r 4. C Solving for the value of r, we get 625 r 4 or r 5. We obtained two values of r, so we have two geometric sequences. D If r 5, the geometric means are E a2 5 5 25, a3 5 5 125, a4 5 5 625. 1 2 3 EP Thus, the sequence is 5, 25, 125, 625, 3125. If r 5, then the geometric means are a2 5 5 25, a3 5 5 125, a4 5 5 625. 1 2 3 D Thus, the sequence is 5, 25, 125, 625, 3125. 30 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. At this point, you already know some essential ideas about geometric sequences. Now, we will learn how to find the sum of the first n terms of a geometric sequence. Do Activity 6. Activity 6: Do the following with a partner. Part 1: PY Consider the geometric sequence 3, 6, 12, 24, 48, 96,... What is the sum of the first 5 terms? There is another method to get the sum of the first 5 terms. Let S5 3 6 12 24 48. O C Multiplying both sides by the common ratio 2, we get 2S5 6 12 24 48 96 D Subtracting 2S5 from S5 , we have E S5 3 6 12 24 48 EP 2S 5 6 12 24 48 96 S5 3 96 D S5 93 S5 93 Try the method for the sequence 81, 27, 9, 3, 1,... and find the sum of the first 4 terms. From the activity, we can derive a formula for the sum of the first n terms, Sn , of a geometric sequence. 31 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Consider the sum of the first n terms of a geometric sequence: Sn a1 a1r a1r 2 ... a1r n 1 (equation 1) Multiplying both sides of equation 1 by the common ratio r, we get rSn a1r a1r 2 a1r 3 ... a1r n 1 a1r n (equation 2) Subtracting equation 2 from equation 1, we get Sn a1 a1r a1r 2 ... a1r n 1 equation 1 rS n a1r a1r 2 ... a1r n 1 a1r n equation 2 PY __________________________________________ Sn rSn a1 a1r n Factoring both sides of the resulting equation, we get O Sn 1 r a1 1 r n . C Dividing both sides by 1 r , where 1 r 0, we get Sn a1 1 r n , r 1. 1 r D Note that since an a1r n 1, if we multiply both sides by r we get E an r a1r n 1 r or an r a1r n. EP Since Sn a1 1 r n 1 1 , a arn 1 r 1 r D Then replacing a1r n by anr, we have a1 an r Sn , r 1. 1 r What if r 1? If r 1, then the formula above is not applicable. Instead, Sn a1 a1 1 a1 1 ... a1 1 2 n 1 a1 a1 a1 ... a1 na1. n terms 32 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Example: What is the sum of the first 10 terms of 2 2 2 ...? Solution: 2 2 2 2 2 2 2 2 2 2 10 2 20 What if r 1? If r 1 and n is even, then Sn a1 a1 1 a1 1 a1 1 ... a1 1 2 3 n 1 PY a1 a1 a1 a1 ... a1 a1 a1 a1 a1 a1 ... a1 a1 0 O However, if r 1 and n is odd, then C Sn a1 a1 1 a1 1 a1 1 ... a1 1 2 3 n 1 a1 a1 a1 a1 ... a1 a1 a1 a1 a1 a1 a1 ... a1 a1 a1 D a1 E a1 1 r n or a1 an r , if r 1 To summarize, Sn 1 r EP 1 r na1, if r 1 In particular, if r 1, the sum Sn simplifies to D 0 if n is even Sn a1 if n is odd Example 1: What is the sum of the first 10 terms of 2 2 2 2 ...? Solution: Since r 1 and n is even, then the sum is 0. Example 2: What is the sum of the first 11 terms of 2 2 2 2 ...? Solution: Since r 1 and n is odd, then the sum is 2. 33 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. Example 3: What is the sum of the first five terms of 3, 6, 12, 24, 48, 96,...? Solution: Since a1 3, r 2, and n 5, then the sum is S5 3 1 25 3 31 93. 1 2 1 a1 an r Alternative Solution: Using Sn , let a1 3, a5 48, and r 2. Then 1 r 3 48 2 3 96 93 S5 93. 1 2 1 1 Part 2: PY Is it possible to get the sum of an infinite number of terms in a geometric sequence? 1 1 1 1 Consider the infinite geometric sequence , , , ,... O 2 4 8 16 If we use the formula Sn a1 1 r n , then C 1 r 1 1 1 1 1 n n 1 2 2 2 2 2 1 1 1 n 1 n D Sn 2 1 . 1 1 2 2 2 2 1 2 2 E The first five values of Sn are shown in the table below. EP n 1 2 3 4 5 Sn 1 3 7 15 31 2 4 8 16 32 D What happens to the value of Sn as n gets larger and larger? Observe that Sn approaches 1 as n increases, and we say that S = 1. To illustrate further that the sum of the given sequence is 1, let us 1 1 1 1 show the sum of the sequence ... on a number line, adding 2 4 8 16 one term at a time: 34 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015. 1 1 1 1 What does this tell us? Clearly, ... 1. 2 4 8 16 We call the sum that we got as the sum to infinity. Note that the 1 common ratio in the sequence is , which is between –1 and 1. We will now 2 derive the formula for the sum to infinity when 1 r 1.
