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Mathematics

Quarter 1 – Module 6 : Geometric

Sequence vs. Arithmetic

Sequence

Department of Education Republic of the Philippines

Mathematics – Grade 10 Alternative Delivery Mode Quarter 1 – Module 6: Differentiates a geometric sequence from an arithmetic sequence. - M10AL-Id- 2 First Edition, 2019 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed material (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 form 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 Assistant Secretary: Alma Ruby C. Torio 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] Development Team of the Learner’s Material Author’s Name: Jess Mitts M. Belingon Reviewers: Eunice Ann B. Puguon, Florentina D. Lunag, Mary Maris B. Tuguinay, Bryan A. Hidalgo Editors: Efiginia B. In-uyay, Genevieve C. Cabbigat Illustrator and Layout Artists: Jess Mitts M. Belingon Management Team: May B. Eclar, Ph.D. CESO V - Regional Director Benedicta B. Gamatero, Assistant Schools Division Superintendent OIC- Schools Division Superintendent Marciana M. Aydinan, PhD, Chief- CID Carmel F. Meris – Regional Chief Education Supervisor - CLMD Ethielyn Taqued, EdD, Regional EPS-LRMDS Edgar H. Madlaing, Regional EPS – AP/ADM Focal Person Lydia I. Belingon, Division EPS-Math/ADM Focal Person

Introductory Message For the facilitator: As the facilitator of this module, please be patient in guiding the learner as he/she undergo this module. Always remind the learner the value of patience and perseverance. Do not forget to remind the learner to use separate sheets in answering all of the activities found in this module. For the learner: I hope that this module will be of big help in completing your Mathematics 10. This is designed to provide you with interactive tasks to further develop the desired learning competencies prescribed in the K-12 Curriculum of the Department of Education. After undergoing this module, you are expected to accurately differentiate a geometric sequence from an arithmetic sequence. 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 contains the learning objectives which you need to accomplish What I know This assesses what you know about the lesson you are to tackle What’s In This connects the current lessons with the previous lessons What’s New This introduces the lesson through an activity What Is It This contains a brief discussion of the lessons What’s More These are activities to check your understanding of the lesson What I have Learned This summarizes the important ideas presented in the lesson What I Can Do This is a real life application of what you have learned Assessment This is a post assessment of what you have learned Additional Activities This is an activity that will strengthen your knowledge about the lesson

This module was designed and written with you in mind. It is here to help you differentiate arithmetic sequence from geometric sequence. The scope of this module permits it to be used in many different learning situations. The lessons are arranged to follow the standard sequence of the course but the pacing in which you read and answer this module will depend on your ability. After going through this module, the learner should be able to: a. determine whether a sequence is geometric or arithmetic. b. explain inductively the difference between arithmetic sequence and geometric sequence using real life-situations. Multiple Choice. Read and analyze the following items and determine the letter of the correct answer from the given choices. Write your answer on a separate sheet of paper. _____ 1. What operation is usually used in obtaining the next term in a geometric sequence? A. Addition C. Division B. Multiplication D. Subtraction _____ 2. What operation is usually used in obtaining the next terms in an arithmetic sequence? A. Addition C. Division B. Multiplication D. Subtraction _____ 3. Which of the following sequences is arithmetic? A. 12, - 12, - 12, - 12 C. 12,

B. 32, 8, 2,

D. 1, 2, 3, 5, 8

_____ 4. Which of the following sequences is geometric? A. 12, - 12, - 12, - 12 C. 12,

What I need to Know What I Know

It was discussed to you earlier that sequences are arrangement of objects or terms which follow a certain pattern. Look around you, can you cite situations, figures, or even routines which follow a certain pattern? Before going to school, are their sequenced routines that you follow? When performing something, do you follow sequential actions? These are just situations where we can see the applications of sequences. Every individual has his or her own choices and decisions. Each follows a different sequence or a pattern as a result of his choices and decisions. Just like us, humans, different sets of numbers follow different sequences or patterns. HOW TO GENERATE SEQUENCES? Determine the pattern being followed by the terms or numbers in each of the following sequences.

1. 1, 2, 3, 4, 5, … ______________________
2. 7, 14, 28, 56, … ______________________
3. 12, 6, 3,

, … ______________________

4. 7, 12, 17, 22, … ______________________

5. 13, 7, 1, - 5 , … ______________________

If you are going to observe the given sequences, there is always a constant value that is being added or multiplied to obtain the terms in the sequence. Let us discover these concepts through this module. Lesson 1 Arithmetic Sequences vs. Geometric Sequences What’s In What’s New

6 Look at the two sequences below, what can you say about the pattern that is being followed by the terms or numbers in each sequence? What is the pattern you observed in Sequence 1? How about in Sequence 2? In the first example, 5 is the first term or denoted as a 1 , then followed by the second term (a 2 ) 12, third term (a 3 ) 19, and so on. You can observe that in order to obtain the second term (a 2 ), you add 7 to the first term (a 1 ), and to obtain the third term (a 3 ), you add 7 to the second term. So, the pattern is 7 is being added to the preceding terms to obtain the next terms. The number that is being added is referred to as the common difference(d). In the second example, 3 is the first term and you can see that 12 is the second term (a 2 ) this was obtained by multiplying 4 to a 1. Multiplying 4 again to a 2 will give you the third term (a 3 ), which is 48, and so on. So, the pattern is 4 is being multiplied to the preceding terms to obtain the next terms. The number that is being multiplied is the common ratio.

Common ratio (r)

Common difference (d)

The common difference can be obtained by : 𝒂𝟐 − 𝒂𝟏or 𝒂𝟐 − 𝒂𝟏 or 𝒂𝟐 − 𝒂𝟏 or Succeeding term – preceding term

su

An arithmetic sequence is a sequence obtained by adding a common difference (d) to the preceding term in order to obtain the next term

3 x4, 12x4, 48

x

What is It The common ratio can be obtained by : 𝒂𝟐 𝒂𝟏 or

or

Let us do the same for examples 2 to 4 to find out if the sequence is arithmetic or geometric. Example 2.) 10, 5, 0, - 5,… 𝑎 2 𝑎 1

5 10

1 2

and

𝑎 3 𝑎 2

0 5

a 2 – a 1 = 5 – 10 = - 5 and a 3 – a 2 = 0 – 5 = - 5

Example 3.) 5, 15, 45,… 𝑎 2 𝑎 1

15 5

= 3 and

𝑎 3 𝑎 2

45 15

Example 4.) 24, 12, 6, 3,… 𝑎 2 𝑎 1

12 24

1 2

and

𝑎 3 𝑎 2

6 12

1 2 ½ is not equal to 0. Thus, sequence is not geometric There is a common difference of – 5. Therefore, the sequence is

arithmetic

There is a common difference of 3.

Therefore, the sequence is geometric

There is a common difference of 𝟏 𝟐

Therefore, the sequence is geometric

ASSESSMENT 1. WHICH IS WHICH?

Directions: Determine if the following sequence is arithmetic, geometric or neither. Write JUST for arithmetic, DO for geometric and IT for neither. Write your answer on the box above the sequence. ASSESSMENT 2. THEY NEED HELP Instructions: Read and analyze the situation carefully. Then, provide what is being asked. SITUATION: Your classmates Ruben and Nathaniel are arguing during your Math subject. You found out that they are arguing about the examples of sequences presented by your teacher. A. Saving 5 pesos on your piggy bank daily. B. An amount of money deposited in a bank that is compounded yearly at an interest rate of 6 %.

1.) a, ar, ar^2 , ar^3 ,…

A. 12, 4,

… B. 13,

SEE THE DIFFERENCE!

What is the main difference that you observed between the two sequences below?

---

---

---

---

____________________________________. INVESTIGATE AND CONCLUDE Read and analyze the situation below. Consider yourself depositing an amount of money in a local bank which offers 5% interest rate compounded annually. If you deposited P10, on that bank, how much will you earn after 3 years?

1. What kind of sequence is illustrated in the above situation?
2. If it shows an arithmetic sequence, then what is the common difference? On the other hand, if it shows a geometric sequence, then what is the common ratio? Provide a justification for your answer using any method. What I have learned What I can do

Read and analyze the following items and determine the letter of the correct answer from the given choices. Write your answer on a separate sheet of paper. _____ 1. What kind of sequence is 7, 20, 33 , …? A. Arithmetic C. Harmonic B. Geometric D. Fibonacci _____ 2. 14 , 7,

… is an example of ____________ sequence. A. Arithmetic C. Harmonic B. Geometric D. Fibonacci For items 3 – 5, refer to the situation below. This Christmas season, Cesar is planning to buy a new mobile phone worth P 10 , 000.00. He started saving P 2 00 pesos on the first month of the year and on the second month, he increases his savings by half of his initial saving. _____ 3. If he continues to follow the same pattern in the succeeding months, what will be his total savings at the end of the year? A. P7, 688. 67 C. P18, 700. 00 B. P17, 299. 51 D. P11, 533. 00 _____ 4. What is the common ratio of the sequence obtained from his savings? A.

C.

B.

D.

_____ 5. When will he be capable of buying the mobile phone? A. at the end of September C. at the end of November B. at the end of August D. at the end of October For Items 6 – 15 , determine whether the given situation is arithmetic, geometric or neither. Write A for arithmetic, G for geometric and N for neither. _____ 6. Half – life of a carbon or any element _____ 7. An amount of money which have a compound interest rate of 10 %. _____ 8. Saving a money on your piggy bank 10 pesos weekly. _____ 9. Each minute hand of the clock that the second hand covers is 5 seconds. Assessment

WHAT’S MORE :ASSESSMENT 1 DO 1. DO 2. IT 3. DO 4. IT 5. JUST 6. JUST 7. JUST 8. DO 9. DO 10. WHAT I CAN DO GEOMETRIC 1. Common Ratio = 21/20 or 1. 05 2. 10,00 pesos which is The initial deposit is 3. compounded annually with a rate of 5 %. To obtain the second term, multiply the initial deposit by 5 %. Then the obtained value is the second term, so, divide the second term by the first term to get the common ratio. , 500 / 10, 000 Common ratio = 10

ASSESSMENT - POST

**11. G A 1.

12. A B 2.
13. G B 3.
14. G C 4.
15. G C 5.** G 6. G 7. A 8. A 9. A 10. WHAT’S NEW Add 1 1. Multiply 2 2. Multiply 3. 𝟏

Add 5 4. 6 – Add 5.

TEST – PRE B 1. A 2. C 3. B 4. A 5. G 6. A 7. G 8. A 9. G 10. G 11.

A 12.

A 13. G 14. G 15. WHAT I HAVE LEARNED 12, 4, 4/3, 4/9, 4/27…is a geometric sequence with a common ratio of 1/3. is an arithmetic 13, 27/2 , 14, 29/2 , 15 … sequence with a common difference of ½. ASSESSMENT 2 WHAT’S MORE : RUBEN 1. Saving 5 pesos on a piggy bank daily shows a 2. constant number that is being added on the preceding days, therefore it shows an arithmetic sequence. While, depositing a money on a bank acquires an interest depending on a constant value ed to the preceding amount, that is being multipli therefore a geometric sequence is obtained.