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Mathematics

Quarter I – Module 5 :

Solving Problems Involving Sequences

M10AL-If- 2

Mathematics – Grade 10 Alternative Delivery Mode Quarter 1 – Module 5 : Solving Problems Involving Sequences First Edition, 2020 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary : Leonor Magtolis Briones Undersecretary : Annalyn M. Sevilla Department of Education, Schools Division of Bulacan Curriculum Implementation Division Learning Resource Management and Development System (LRMDS) Capitol Compound, Guinhawa St., City of Malolos, Bulacan Email address: [email protected] Development Team of the Module Author : Jeffrey B. Gonzales Language Reviewer : Grace M. Yumang, PhD Content Editor : Richelle SD. Sagum Illustrator : Jeffrey B. Gonzales Layout Artist : Jeffrey B. Gonzales Management Team: Gregorio C. Quinto, Jr., EdD Chief, Curriculum Implementation Division Rainelda M. Blanco, PhD Education Program Supervisor - LRMDS Agnes R. Bernardo, PhD EPS-Division ADM Coordinator Glenda S. Constantino Project Development Officer II Francisco B. Macale Mathematics - Division Focal Person Joannarie C. Garcia Librarian II

For the Learner: Welcome to the Math 10 Module on Solving Problems Involving Sequences. This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner. In addition to the material in the main text, you will also see this box in the body of the module: For the Facilitator: Welcome to the Math 10 Module on Solving Problems Involving Sequences. This module was collaboratively designed, developed and reviewed by educators both from 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. This learning resource hopes to engage the learners into guided and independent learning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills while taking into consideration their needs and circumstances. As a facilitator, you are expected to orient the learners on how to use this module. You also need to keep track of the learners’ progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and the learners as they do the tasks included in the module. This module has the following parts and corresponding icons: This will give you an idea of the skills or competencies you are expected to learn in the module. This part includes an activity that aims to check what you already know about the lesson to take. If you get all the answers correctly (100%), you may decide to skip this module.

Introductory Message

Notes to the Teacher The activities in this module are arranged from simple to complex to help the learner gradually master the desired learning competency. Give him/her the needed support and guidance so that he/she will be able to perform the tasks to prepare him/her later on in solving problems involving sequences.

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

1. Use the module with care. Do not put unnecessary mark/s on any part of the module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it. If you encounter any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator. Always bear in mind that you are not alone. We hope that through this material, you will experience meaningful learning and gain deep understanding of the relevant competencies. You can do it!

______ 6. How much will it cost a 14 - kilometer trip? a. 141 pesos b. 149 pesos c. 157 pesos d. 165 pesos ______ 7. If the additional charge for every kilometer is 9 pesos, how many kilometers can be travelled for a 135-peso budget? a. 10 kilometers b. 11 kilometers c. 12 kilometers d. 13 kilometers For items 8 to 9 : Kristine conducted an experiment and noticed that when she tosses a coin once, 2 possible outcomes can occur. She tossed the coin twice and there were 4 possible outcomes while 8 possible outcomes occurred when she tossed the coin three times. ______ 8. If the coin was tossed for 5 times, how many possible outcomes will there be? a. 32 b. 64 c. 128 d. 256 ______ 9. How many tosses are needed to make 512 possible outcomes? a. 9 b. 10 c. 11 d. 12 ______ 10. France used her savings to buy a new car amounting to Php720,000.00. Suppose a car’s value depreciates yearly at 10%, what is her car’s value after 4 years? a. 425,152.80 b. 472,392.00 c. 524,880.00 d. Php583,200. ______ 11. A country struck by the Corona Virus Disease – 19 (CoViD19) reached its peak infecting 81,920 people in total. It was found that the virus spread is in a Geometric Progression and that the number of people infected gets doubled each day reason for its government to implement a total lockdown. If there were only 5 people infected on the first day, how many days had passed to reach its peak? a. 21 days b. 18 days c. 15 days d. 11 days For items 12 to 13: During a free-fall, a skydiver jumps 16 feet, 48 feet, and 80 feet on the first, second, and third fall, respectively. He continues to jump at this rate. ______ 12. How many feet will he have to jump on the 10th^ fall? a. 304 b. 336 c. 3149 28 d. 944784 ______ 13. How many feet will he have to jump on the 18th^ fall? a. 496 b. 528 c. 560 d. 592 For items 14 to 15: A new square is formed by joining the midpoints of the consecutive sides of a square 8 inches on a side. ______ 14. If the process continued until there are already six squares, find the sum of the areas of all squares in square inches. a. 96 b. 112 c. 124 d. 126 ______ 15. If the process continued until there are already five squares, find the sum of the areas of all squares in square inches. a. 96 b. 112 c. 124 d. 126

From the previous modules, you were able to encounter number patterns called Arithmetic and Geometric Sequence. Let us have a simple recall. What is an arithmetic sequence? What is a geometric sequence? How are these sequences formed? Let us find out more from the details below. Arithmetic Sequence is a sequence of numbers that follows a pattern. The terms after the first are obtained by adding the preceding term a non-zero constant called the common difference ( d ). Take a look at this sequence below.

Remember Me! Directions: Tell whether each of the following sequences is arithmetic or not. Shade the sequence if it is arithmetic , otherwise leave the figure blank.

1. 3, 10, 17, 24, 31, … 6. 3, 9, 27, 81, 243, …
   2. • 8 , - 16 , - 3 2, - 64 , - 128 , … 7. ½, 1/4, 1/8, 1/16, …
   3. 9, 3, - 3, - 9, - 15, … 8. 5, 25, 125, 625, …
   4. 3 , - 3 , 3 , - 3 , 3 , … 9. 12, 15, 18, 21, 24, …
   5. 2 , 12 , 22 , 32 , 42, … 10. 3, 0, 0, 0, 0 Geometric Sequence is a sequence of numbers that follows a pattern. The terms after the first are obtained by multiplying the preceding term a non-zero constant called the common ratio ( r ). Take a look at this sequence below.

1 st^ term a 1 2 nd^ term 3 rd^ term 4 th^ term 5 th^ term 6 th^ term a 2 , a 3 , a 4 , a 5 , a 6 2 2 2 2 2 The sequence 2, 4, 8, 16, 32, 64 is a geometric sequence whose common ratio (r) is 2. The common ratio is obtained by dividing any term by the term that precedes it. Example: 64 ÷ 32 = 2 32 ÷ 16 = 2 16 ÷ 8 = 2 In this case, 2 is the common ratio. 1 st^ term a 1 2 nd^ term 3 rd^ term 4 th^ term 5 th^ term 6 th^ term a 2 , a 3 , a 4 , a 5 , a 6 3 3 3 3 3 The sequence 4, 7, 10, 13, 16, 19 is an arithmetic sequence whose common difference (d) is 3. The common difference is obtained by subtracting any term by the term that precedes it. Example: 19 - 16 = 3 16 - 13 = 3 13 - 10 = 3 In this case, 3 is the common difference.

GEOMETRIC SEQUENCE RULE

5 , 10 , 20 , 40 , 80 , 160 , … ✓ The first term is 5 or a 1 = 5 ✓ The common ratio is 2 or r = 2 From the rule above, we can say that each term of a geometric sequence is obtained using the formula:

a n = a 1 ● r n-^1

where: a n = nth^ term (the nth^ term refers to the location or position of the term in the sequence) a 1 = first term (the first term refers to the first number entry in the sequence) r = common ratio (the common ratio refers to the same numbers obtained by dividing any term by the term that precedes it) Example:

1. What is the 7th^ term of the geometric sequence 3, 6, 12, 24, 48, …? Given: a 1 = 3, r = 2, n = 7 Solution: an = a 1 ● r n-^1 a 7 = 3 ● 27 -^1 a 7 = 3 ● 26 a 7 = 3 ● 64 a 7 = 192
2. What is the 8th^ term of the geometric sequence 2, - 4 , 8 , - 16 , 32 , …? Given: a 1 = 2 , r = - 2 , n = 10 Solution: an = a 1 ● r n-^1 a 10 = 2 ● (-2)^10 -^1 a 10 = 2 ● (-2)^9 a 10 = 2 ● - 512 a 10 = - 1024 Term Other Ways to Write the Term Factored Form Exponential Form

a 1 = 5 a 1 = 5 a 1 = 5 x 2^0

a 2 = 10 a 2 = 5 x 2 a 2 = 5 x 2^1

a 3 = 20 a 3 = 5 x 2 x 2 a 3 = 5 x 2^2

a 4 = 40 a 4 = 5 x 2 x 2 x 2 a 4 = 5 x 2^3

a 5 = 80 a 5 = 5 x 2 x 2 x 2 x 2 a 5 = 5 x 2^4

a 6 = 160 a 6 = 5 x 2 x 2 x 2 x 2 x 2 a 6 = 5 x 2^5

a n =? ---------------------------^ a n = a 1 x r n-^1

a (^) 6 = 5 x 2 5 Subtract 1

a^1 r

Sn = Sn =

ARITHMETIC SERIES

Arithmetic Series – refers to the sum of the terms of an arithmetic sequence. We use the formula: n[ a 1 +(a 1 + (n-1)d) ] or n 2 2 Example: Find the sum of the first 20 terms of the arithmetic sequence - 2, - 5, - 8, - 11, … Given: a 1 = - 2, d = - 3, n = 20 Solution: n[ a 1 +(a 1 + (n-1)d) ] 2 20 [ - 2 +(- 2 + ( 20 - 1)- 3 )] 2 20 [ - 2 +(- 2 + ( 19 )- 3 )] 2 20 [ - 2 +(- 2 + - 57 )] 2 20( - 2 + - 59) 2 20(-61) 2

• 1220 2 - 610

GEOMETRIC SERIES

Geometric Series – refers to the sum of the terms in a geometric sequence. a 1 (1-rn) (1-r) a 1 - anr (1-r) Sn = n(a 1 ) Sn = 0 Sn = Sn = [2a^1 + (n-1) d] Sn = Sn = Sn = Sn = Sn = Sn = Sn = Sn = a. Geometric Series Formula *Use this formula only when the common ratio is ≠ 1 and the last term is unknown. b. Geometric Series Formula *Use this formula only when the common ratio is ≠ 1 and the last term is given. c. Geometric Series Formula *Use this formula only when the common ratio is = 1 d. Geometric Series Formula *Use this formula only when the common ratio is = - 1 and n is even.

Sn = S 8 = S 8 = S 8 = S 8 = Substitute the values of n , r, a 1 , an then compute for the sum. The sum of the first 8 terms of the geometric sequence 3, - 6, 12, - 24 , …, is – 255. The sum of the first 15 terms of the sequence

• 3, 3, - 3, 3, …, is - 3. a 1 (1-rn) (1-r) 3(1-(-2)^8 ) (1- - 2) 3(1-(-2)^8 ) 3 3 (1-256) 3 3(-255) 3 S 8 = - 255

3. In the sequence - 3, 3, - 3, 3, … , the common ratio (r ) = - 1. Since you are asked to give the sum of the first 15 terms, n = 15. We will use the formula Sn = a 1 because the number of terms is an odd number. Given: a 1 = - 3 n = 15 Sn = a 1 S 15 = - 3 You were able to recall the arithmetic and geometric sequences. Let us have another activity! You may ask a friend to help you perform the next activity. Find a Solution! Directions: Using the formula in finding the nth^ term of an arithmetic and geometric sequence, solve the following problems.
4. A side of an apartment building is shaped like a steep staircase. The windows are arranged in columns. The first column has 2 windows, the second column has 5 windows, the next has 8, then 11, and so on. How many windows are on the side of the apartment if there are 15 columns?
5. Assuming that the geometric sequence continues, what is the height of a bouncing ball on its 8th^ bounce if its initial height after the first drop is 140 ft., next is 70 ft., and so on?

If your answer on the problems are 44 and 35/32 respectively, you are right! What you did was to apply the concepts of arithmetic and geometric sequence in a given worded problem. SOLVING PROBLEMS INVOLVING SEQUENCES From here, you will need to apply what you have learned from the previous activity. Let me guide you as you go through the module. Are you ready? Let’s begin. Have a look on these examples: A theater in Pasay City has 27 seats in the first row of its center section. Each row behind the first gains additional 4 seats. How many seats are in the ninth row of the center section? The problem above is an arithmetic sequence. After every row, you are adding 4 seats behind. It tells you that the sequence would be 27, 31, 35, 39, … Since it is asking for the number of seats on the nineth row, then we could get the following given: a 1 = 21 n = 9 d = 4 We will use the formula,

an = a 1 + (n-1)d

Solution: an = a 1 + (n-1)d a 9 = 27 + ( 9 - 1) 4 Therefore, the are 59 chairs on the nineth row of the a 9 = 27 + ( 8 ) 4 center section. a 9 = 27 + 32 a 9 = 59 A research laboratory is to begin experimentation with bacteria that doubles every 4 hours. The lab starts with 200 bacteria. How many bacteria will there be in total after 24 hours? The problem above is a geometric sequence. After every 4hours, the bacteria doubles. It tells you that the sequence would be 200, 400, 800, … Since it is asking for the total amount of bacteria after 24 hours, then we could get the following given: a 1 = 2 00 n = 6 (since the bacteria doubles only after every 4 hours, then n = 6) r = 2

2. The Summer Olympics happen every four years. Suppose the first ever Summer Olympics was held last 2012, in which year will the 14th^ Summer Olympics occur? Is the sequence arithmetic or geometric?__________________________ What are the given? ___________ ___________ ___________ Formula: ___________________ Solution: Final Answer: _____________
3. Atom stacks his lego pieces to create columns with a ladder-like pattern. He started with 3 lego pieces on the first column and every succeeding column contains 5 more lego pieces than the previous column. How many lego pieces are needed to be stacked on the 12th column? Is the sequence arithmetic or geometric?__________________________ What are the given? ___________ ___________ ___________ Formula: ___________________ Solution: Final Answer: _____________

You did an amazing job in the previous activities. You are now ready to take the next challenge.

Independent Activity 1 :

It’s my Type! Directions: Determine whether the given situations/problems are arithmetic or geometric. Put a check mark ( ) on the circle that corresponds to your answer. Ms. Caballero wants to conduct a study about the Proficiency Level in Math of the Grade 10 students of Vedasto R. Santiago High School. She wanted to be fair on the selection of respondents, so she used a probability sampling technique to get the desired number of respondents. The respondents were obtained using the numbers that are multiples of five. Arithmetic Geometric A teacher wants to determine the seating capacity of a gymnasium with 18 seats on the first row, 20 on the second, 22 on the third. The

sequence of the seating capacity per row continues until the 14th^ row.

Arithmetic Geometric Tomas bought a car amounting to 600,000 pesos. He learns that the car’s value decreases each year passing by. He was surprised when he computed his car’s value after 5 years with a depreciation rate of 8%. Arithmetic Geometric A ball was dropped and bounces back half the previous height. The initial height where the ball is dropped is 320 feet and continues to follow the same sequence until it stops. Arithmetic Geometric Your father wants you to help him build a shed in the backyard. He says he will pay you 20 pesos for the first week and add an additional 6 pesos each week thereafter. The project will take 5 weeks. Arithmetic Geometric

Independent Activity 2:

Bring it On! Directions: Using the arithmetic and geometric sequences formula, solve the following problems. Write your answer on the box provided.

1. How many student participants will there be on the 5th^ day?

2. How many student participants will there be on the 9th^ day?

3. How many student participants will there be on the 7th^ day?

4. How many student participants will there be on the 10th^ day?

5. Suppose the same sequence is followed. In which day will the student – participants

be equal to 1 18? A 10 – day series of talk was conducted at the gymnasium of Bajet-Castillo National High School. It was attended by 20 students on the first day. Finding the talk timely and interesting, these 20 students shared to 14 students and 14 more students came on the second day, another 14 more students came on the third day, and so on.

Independent Assessment 2 :

Remember “Sum”thing? Directions: Compute the sum of the terms of the given geometric sequence below. Apply the geometric series formula to answer the given items. * A scientific calculator may be used.

1. A doctor discovered that when a chemical is poured on a certain solution where bacteria grows, the number of bacteria present in it is reduced by half. Suppose the initial bacteria on the container is 2560 when the chemical is poured, find the sum of the remaining bacteria after
2. A math teacher asks the students to subtract and add 12 to 6 simultaneously to form a sequence. Determine the sequence formed and give the sum of the
3. When a coin is tossed once, the possible outcomes is 2, 4 possible outcomes when tossed twice, 8 possible outcomes when tossed thrice, and so on. Find the sum of the terms formed. Sum More! Directions: Answer the following word problem involving geometric series. Write your answer on the given space. A microbiologist was able to isolate 12 endophytic bacterial strains from nipa palm in Bulacan, Philippines. In this study, screening of endophytic bacteria examined their multiple plant growth-promoting traits. These bacteria are able to increase their numbers by the process of replication called binary fission whereby their population doubles every generation time. Find the total population (sum) of bacteria after: a. 9 generation time. b. 7 generation time. 1. 5 terms. ____________ 2. 7 terms. ____________ 3. 10 terms. ____________ 4. first 11 terms. ____________ 5. first 14 terms. ____________ 6. first 10 terms. ____________ 7. 10 terms ____________ 8. 7 terms ____________ 9. 12 terms ____________ 10. infinite terms ____________ 11.