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# Geometric Sequence by Alma Baja

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### Geometric Sequence by Alma Baja

1. 1. STRATEGIC INTERVENTION MATERIAL MATHEMATICS 10 PREPARED BY: ALMA EDQUID BAJA T-2 Department of Education Region III Schools Division of Zambales GUISGUIS NATIONAL HIGH SCHOOL Guisguis, Sta. Cruz, Zambales
2. 2. LEAST MASTERED SKILLS Illustrates Geometric Sequence Sub-Tasks: • Distinguish between arithmetic and geometric sequences. • Find the common ratio of a geometric sequence. • Determine the geometric means and nth term of a geometric sequence. • Find the sum of the terms of a given finite and infinite geometric sequence.
3. 3. A Geometric Sequence is sequence where each term after the first is obtained by multiplying the preceding term by a nonzero constant called the common ratio. How are sequences used to model and solve many mathematical ideas and real- life situations? You need the concept of ratio in order to understand the geometric sequences. We will explore that sequence by doing some activities. Are you ready? Lets go!
4. 4. Activity #01 Divide and Conquer Find the ratio of the second number to the first number. 1. 2, 8 6. -49, 7 2. -3, 9 7. ¼, ½ 3. 1, ½ 8. 𝒂 𝟐 , 𝒂 𝟑 4. -5, -10 9. k – 1, k 5. 12, 4 10. 3m, 3mr
5. 5. Activity #02 Give Me My Ratio! Identify the common ratio of the following geometric sequences. 1. 4, 12, 36, . . . 2. 3, 6, 12, 24, . . . 3. 36, 18, 9, . . . 4. -3, 12, -48, . . . 5. -5, -25, -125, . . . Thus, in the geometric sequence 2, 4, 8, 16, 32, . . . , the common ratio is 2 since 𝟑𝟐 𝟏𝟔 = 2
6. 6. The next activity will test whether you can identify geometric sequences or not. Goodluck! Activity #03: I’ll Tell You What You Are 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, . . . 5. 4, 0, 0, 0, 0, . . .
7. 7. Activity # 04: Missing You Find the missing terms in each geometric sequence. 1. 3, 12, 48, ___, ___ 2. ___, ___, 32, 64, 128, . . 3. 120, 60, 30, ___, ___ 4. 5, ___, 20, 40, ___, 5. ___, 4, 12, 36, ___
8. 8. Remember: The nth term of a geometric sequence is 𝒂 𝒏 = 𝒂 𝟏 𝒓 𝒏−𝟏 where 𝒂 𝟏 = the first term 𝒂 𝒏 = the nth term r = the common ratio Find the tenth term of the geometric sequence 2, 4, 8, . . . Solution: 𝑎1 = 2; r = 2; n = 10 Formula: 𝑎 𝑛= 𝑎1 𝑟 𝑛−1 𝑎10= 2(2)10−1 𝑎10 = 2(2)9 = 1024 – the tenth term Activity #05: There’s More on Geometric Sequences Use the nth term of a geometric sequence 𝒂 𝒏 = 𝒂 𝟏 𝒓 𝒏−𝟏 to answer the following questions. 1. What is the 5th term of the geometric sequence 3, 6, 12, . . .? 2. Find the 6th term of a geometric sequence where the 2nd term is 6 and the common ratio is 2. 3. Find the tenth term of the geometric sequence 2, -6, 18, . . .
9. 9. Activity #06 Finding Geometric Means 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 sequences the Geometric Means. Example: Insert 2 geometric means between 5 and 625 Solution: Let 𝑎1= 5 and 𝑎4 = 625. we will insert 𝑎2, 𝑎𝑛𝑑 𝑎3. Since 𝑎4 = 𝑎1 𝑟3 , then 625 = 5𝑟3 . Solving for the value of r, we get 125 = 𝑟3 or r = ±5 We obtained two values of r, so we have two geometric sequences.
10. 10. If r = 5, the geometric means are 𝑎2= 5 5 1 = 25 𝑎3= 5 5 2 = 125. Thus, the sequence is 5, 25, 125, 625. If r = -5, the geometric means are 𝑎2= 5 −5 1 = -25 𝑎3= 5 5 2 = 125. Thus, the sequence is 5, -25, 125, -625. Find the indicated number of geometric means between each pair of numbers. 1. 16 and 81 (3) 2. 256 and 1 (3) 3. -32 and 4 (2) 4. 1 3 and 64 3 (1) 5. 2xy and 16𝑥𝑦4 (1) Was knowing the nth term of a geometric sequence helpful in finding geometric means?
11. 11. Activity # 07: Sum of Terms in a Geometric Sequence Answer the following: 1. Find the sum of the first 5 terms of 4, 12, 36, . . . 2. Find the sum of the first 6 terms of 3, -6, 12, -24, . . . 3. . Find the sum of the first 6 terms of -3, 3, -3, 3, . . . 4. Find the sum of the first 7 terms of -3, 3, -3, 3, . . . 5. Find the sum of the first 8 terms of - 3 4 , 3 4 , 3 4 , 3 4 , . . . Formula: Sum of Finite Geometric Sequence 𝑆 𝑛 = 𝑎1(1−𝑟 𝑛) 1−𝑟 or 𝑎1−𝑎 𝑛 𝑟 1−𝑟 where: 𝑆 𝑛 = the sum 𝑎1 = the first term r = the common ratio, r≠1. 𝑆 𝑛 = n 𝑎1 if r = 1 Sum of Infinite Geometric Sequence S = 𝑎1 1−𝑟 where: 𝑆 𝑛 = the sum 𝑎1 = the first term r = the common ratio, /r/ < 1.
12. 12. Assessment #01: How well do you know me? Determine whether each sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, give the common difference; if geometric, give the common ratio. 1. 6, 18, 54, 162, . . . 2. 4, 10, 16, 22, . . . 3. 1, 1, 2, 3, 5, 8, . . . 4. 625, 125, 25, 5, . . . 5. 5, 8, 13, 21, 34, . . .
13. 13. Assessment #02: Do You Remember Me? Answer the following. 1. Find the fifth term of the geometric sequence 5, 10, 20, . . . 2. Find the geometric mean between 3 and 12. 3. Insert two geometric mean between 3 and 81. 4. Find the sum of the first 8 terms of the geometric sequence -5, 5, -5. 5, . . . 5. Find the sum of the first 7 terms of the geometric sequence -5,
14. 14. State whether the given sequence is arithmetic or geometric. Then give the next term of the sequence. 1. 8, 16, 24, 32, . . . 2. 1 3 , 1 9 , 1 27 , 1 81 , . . . 3. 5, 10, 15, 20, . . . 4. 5, 10, 20, 80, . . . 5. 9, 19, 29, 39, . . .Mastery Points! Can You • Distinguish between arithmetic and geometric sequences? • Identify the common ratio of a geometric sequence? • Identify the geometric means and find the nth term of a geometric sequence? • Find the sum of the terms of a given finite and infinite geometric sequence.
15. 15. - Mathematics Learners Module-Grade10 (DEPED) p. 26 - 47 - Intermediate Algebra - Soledad Jose-Dilao, Ed.D. Julieta G. Bernabe, Published by JTW Corporation. - Intermediate Algebra, Corazon Alano, et.al, Abiva Publishing House, Inc. - http://www.goggle.com
16. 16. Activity 1: Divide and Conquer 1. 4 6. -1/7 2. -3 7. 2 3. ½ 8. a 4. 2 9. 𝒌 𝒌−𝟏 5. 1/3 10. r Activity 2: Give Me My Ratio! 1. r = 3 2. r = 2 3. r = ½ 4. r = - 4 5. r = 5 Activity 3: I’ll TellYouWhatYou Are 1.Geometric Sequence 2. Not 3. Geometric Sequence 4. Geometric Sequence 5. Not Activity 4: MissingYou 1. 192, 768 2. 8, 16 3. 15, 𝟏𝟓 𝟐 4. 10, 80 5. 𝟒 𝟑 , 108
17. 17. Activity 5: There’s More On Geometric Sequences 1. 𝒂 𝟓 = 𝟒𝟖 2. 𝒂 𝟔 = 96 3. 𝒂 𝟏𝟎 = −𝟑𝟗𝟑𝟔𝟔 Activity 7: Sum of Terms in a Geometric Sequence 1. 484 2. -63 3. 0 4. -3 5. 6 Activity 6: Finding Geometric Means 1. 24, 36, 54 -24, 36, -54 2. 64, 16, 4 -64, 16, -4 3. 16 and -8 4. 𝟖 𝟑 - 𝟖 𝟑 5. 𝟒𝒙𝒚 𝟐 and 𝟖𝒙𝒚 𝟑
18. 18. Assessment #1 HowWell DoYou Know Me? 1. Geometric r = 3 2. Arithmetic d = 6 3. Neither 4. Geometric r = 1/5 5. Neither 6. Neither 7. Geometric r = ½ 8. Arithmetic d = 2 Assessment #2 DoYou Remember Me? 1. 𝒂 𝟓 = 80 2. 7 𝟏 𝟐 3. 9 and 27 4. 0 5. -5
19. 19. Prepared and Submitted by: ALMA E. BAJA T-2 Approved: JULITA M. VALDEZ Principal II
20. 20. I did great and fully understood the lesson 51 - 67 I did good and partially understood the lesson. I still need to review. 21 - 50 I need to go back with the lesson because there are still areas I need to work on. I will not give up. 1-20 I My Score is ________