Sequences and Series in Mathematics

Delve into the world of sequences and series, understanding the concepts of arithmetic and geometric progressions. Learn to identify finite and infinite sequences, explore various patterns, and model them mathematically. From arithmetic to geometric, finite to infinite, this content offers a comprehensive overview for math enthusiasts.

• Mathematics
• Sequences
• Series
• Arithmetic Progression
• Geometric Progression

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1. Sequences and Series Arithmetic Geometric ?1= first term ??= ?? 1 ? Where do I belong? ,?,?,?, ??=? 2(?1+ ??) ???????? ??????? ?1 ?1= first term ??= ?? 1+ ? ?1= first term ??=?1(1 ??) ???? ???? # ,?,?,?, ? = ?? ? =? + ? 1 ? ??= ?? 1+ ? ??= ?? 1 ? ????? ???? # ??= ?1 ?? 1 ? = ??= ?1+ (? 1) ? ?1= first term 1 ? ??= ?1 ?? 1 2 ??= ?1+ (? 1) ? ??=?1(1 ??) ??=? ???? ????? = ?????? ????????? = ??????????? 1 ? 2(?1+ ??) ?1 ? = ????? ????? = ???? ????? + 1 1 ? ,?,?,?, ? =? + ? ,?,?,?, ? = ?? Pg. 607 #1-32 2

2. Sequence and Series Describe Arithmetic Geometric Summation Modeling 10 10 10 10 10 20 20 20 20 20 30 30 30 30 30 40 40 40 40 40 50 50 50 50 50

3. Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 2, 8, 14, 20, 26 Answer Answer

4. Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 2, 8, 14, 20, 26 Finite Arithmetic Sequence

5. Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 2, 8, 32, 128, Answer Answer

6. Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 2, 8, 32, 128, Infinite Geometric Sequence

7. Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 1 + 5 + 9 + Answer Answer

8. Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 1 + 5 + 9 + Infinite Arithmetic Series

9. Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 16 + 8 + 4 + 2 + 1 Answer Answer

10. Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 16 + 8 + 4 + 2 + 1 Finite Geometric Series

11. Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 3(0.25)? ?=1 Answer Answer

12. Describe each using: Sequence/Series; Arithmetic/Geometric; Finite/Infinite 3(0.25)? ?=1 Infinite Geometric Series

13. Write in summation notation 31, 28, 25, 22,...,2 Answer Answer

14. Write in summation notation 31, 28, 25, 22,...,2 12 ??= ?1+ ? 1 ? 2 = 31 + ? 1 3 2 = 31 + 3? 3 2 = 34 + 3? 36 = 3? 12 = ? 31 + (? 1) 3 ?=1 Or 12 3? 34 ?=1

15. Write a recursive definition, explicit formula and find the 15 term for the sequence: 12, 21, 30, 39, . . . Answer Answer

16. Write a recursive definition, explicit formula and find the 15 term for the sequence: 12, 21, 30, 39, . . . Recursive: ?1= 12; ??= ?? 1+ 9 Explicit: ??= 12 + (? 1) 9 ?15= 12 + 15 1 9 = 138

17. Write a recursive definition, explicit formula and find the 18 term for the sequence: 45, 37, 29, 21, . . . Answer Answer

18. Write a recursive definition, explicit formula and find the 18 term for the sequence: 45, 37, 29, 21, . . . Recursive: ?1= 45; ??= ?? 1 8 Explicit: ??= 45 + (? 1) ( 8) ?18= 45 + 18 1 8 = 91

19. Find the missing term of the arithmetic sequence , 1, ,11,... Answer Answer

20. Find the missing term of the arithmetic sequence , 1, ,11,... = 1 + 11 = 5 2

21. Find the missing terms of the arithmetic sequence , 13, , , ,3,... Answer Answer

22. Find the missing terms of the arithmetic sequence 9, 1, , 13, , , ,3,... 5, = 13 + 3 = 5 2 = 13 + ( 5) = 9 2 = 5 + 4 = 1

23. Write in summation notation 5,10,20,40, ,640 Answer Answer

24. Write in summation notation 5,10,20,40, ,640 ??= ?1 ?? 1 640 = 5 2? 1 128 = 2? 1 ? 1 = log2128 ? 1 = 7 ? = 8 8 5 2? 1 ?=1

25. Write a recursive definition, explicit formula and find the 10 term for the sequence: 3,6, 12,24,... Answer Answer

26. Write a recursive definition, explicit formula and find the 10 term for the sequence: 3,6, 12,24,... Recursive: ?1= 3; ??= ?? 1 2 Explicit: ??= ( 3) ( 2)? 1 ?10= ( 3) ( 2)10 1= 1536

27. Write a recursive definition, explicit formula and find the 9 term for the sequence: 2, 10, 50, 250,... Answer Answer

28. Write a recursive definition, explicit formula and find the 9 term for the sequence: 2, 10, 50, 250,... Recursive: ?1= 2; ??= ?? 1 5 Explicit: ??= ( 2) (5)? 1 ?9= 2 59 1= 781250

29. Find the missing term of the geometric sequence ,3, ,12,... Answer Answer

30. Find the missing term of the geometric sequence ,3, ,12,... = 12 3 = 6

31. Find the missing terms of the geometric sequence , 20, , , , 1.25,... Answer Answer

32. Find the missing terms of the geometric sequence 10, , 1.25,... 2.5, , 20, , , 5, = 20 1.25 = 5 = 20 5 = 10 = 5 1 2= 2.5

33. Evaluate the sum of each series 5 + 11 + 17 + + 35 Answer Answer

34. Evaluate the sum of each series 5 + 11 + 17 + + 35 ??= ?1+ (? 1) ? 35 = 5 + ? 1 6 35 = 5 + 6? 6 35 = 6? 1 36 = 6? 6 = ? ??=? 2(?1+ ??) ?6=6 2(5 + 35) = 120

35. Evaluate the sum of each series 4 + 16 + 64 + + 4096 Answer Answer

36. Evaluate the sum of each series 4 + 16 + 64 + + 4096 ??= ?1 ?? 1 4096 = 4 4? 1 1024 = 4? 1 ? 1 = log41024 ? 1 = 5 ? = 6 ??=?1(1 ??) 1 ? ?6=4(1 46) 1 4 = 5460

37. Evaluate the sum of each series 15 7? 5 ?=4 Answer Answer

38. Evaluate the sum of each series 15 7? 5 ?=4 ??=? 2(?1+ ??) ?4= 7 4 5 = 23 ?15= 7 15 5 = 100 ?12=12 2(23 + 100) = 738

39. Evaluate the sum of each series 7 4(5)? 1 ?=1 Answer Answer

40. Evaluate the sum of each series 7 4(5)? 1 ?=1 ??=?1(1 ??) ?1= 4(5)1 1= 4 1 ? ?6=4(1 57) 1 5 = 78124

41. Evaluate the sum of each series 3 3?2 ?=1 Answer Answer

42. Evaluate the sum of each series 3 3?2 ?=1 ?1= 3(1)2= 3 ?2= 3(2)2= 12 ?3= 3(3)2= 27 + 42

43. The number of toy rockets made by an assembly line for 8 hours forms an arithmetic sequence. If the line produced 40 rockets in hour one and 43 rockets in hour two, how many rockets will be produced in hour seven? How many rockets will be produced in one 8 hour day? Answer Answer

44. The number of toy rockets made by an assembly line for 8 hours forms an arithmetic sequence. If the line produced 40 rockets in hour one and 43 rockets in hour two, how many rockets will be produced in hour seven? How many rockets will be produced in one 8 hour day? ??= ?1+ (? 1) ? ??= 40 + ? 1 3 ??=? ?8=8 2(?1+ ??) 2(40 + 61) = 404 ?7= 40 + 7 1 3 = 58

45. You invested money in a fund and each month you receive a payment for your investment. Over the first four months, you received $50, $57, $64, $71. If this pattern continues, how much will you receive in the 12th month and who much will you receive for the entire year? Write an explicit equation to model the problem. Answer Answer

46. You invested money in a fund and each month you receive a payment for your investment. Over the first four months, you received $50, $57, $64, $71. If this pattern continues, how much will you receive in the 12th month and who much will you receive for the entire year? Write an explicit equation to model the problem. ??=? 2(?1+ ??) ??= ?1+ (? 1) ? ??= 50 + ? 1 7 ?8=12 2(50 + 127) ?12= 50 + 12 1 7 = 127 = 1062

47. You are trying to save $1500. You begin with $5 and save $3 more than the previous week for 30 weeks. Will you meet your goal? Write an explicit formula to model this problem What is the amount you will save in week 30? What is the total amount you will save over 30 weeks? Answer Answer

48. You are trying to save $1500. You begin with $5 and save $3 more than the previous week for 30 weeks. Will you meet your goal? Write an explicit formula to model this problem What is the amount you will save in week 30? What is the total amount you will save over 30 weeks? No ??=? 2(?1+ ??) ??= ?1+ (? 1) ? ??= 5 + ? 1 3 ?30=30 2(5 + 92) = 1455 ?30= 5 + 30 1 3 = 92

49. You saved $500 this year. Each year you plan to save 5% more than the previous year. Write an explicit formula to model this situation. How much will you save in the 8 year? How much will you have saved totally over years, assume you do not spend anything? Answer Answer

50. You saved $500 this year. Each year you plan to save 5% more than the previous year. Write an explicit formula to model this situation. How much will you save in the 8 year? How much will you have saved totally over years, assume you do not spend anything? ??=?1(1 ??) ??= ?1 ?? 1 ??= 500 (1.05)? 1 1 ? ?8=500(1 1.058) 1 1.05 ?8= 500 (1.05)8 1 = 703.55 = 4774.55

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