2. Packet #29 Arithmetic and Geometric Sequences
Math 160
Packet #29
Arithmetic and Geometric Sequences

3. An arithmetic sequence is a sequence of the form: 𝑎, 𝑎+𝑑, 𝑎+2𝑑, 𝑎+3𝑑, … 𝑎 is the first term, and 𝑑 is the common difference. The 𝒏th term is given by 𝑎 𝑛 = ______________

4. An arithmetic sequence is a sequence of the form: 𝑎, 𝑎+𝑑, 𝑎+2𝑑, 𝑎+3𝑑, … 𝑎 is the first term, and 𝑑 is the common difference. The 𝒏th term is given by 𝑎 𝑛 = ______________

5. An arithmetic sequence is a sequence of the form: 𝑎, 𝑎+𝑑, 𝑎+2𝑑, 𝑎+3𝑑, … 𝑎 is the first term, and 𝑑 is the common difference. The 𝒏th term is given by 𝑎 𝑛 = ______________

6. An arithmetic sequence is a sequence of the form: 𝑎, 𝑎+𝑑, 𝑎+2𝑑, 𝑎+3𝑑, … 𝑎 is the first term, and 𝑑 is the common difference. The 𝒏th term is given by 𝑎 𝑛 = ______________
𝒂+ 𝒏−𝟏 𝒅

7. Ex 1. Find the common difference and the 100th term of the following arithmetic sequence. 11, 6, 1, −4, −9, …

8. Ex 1. Find the common difference and the 100th term of the following arithmetic sequence. 11, 6, 1, −4, −9, …

9. Ex 1. Find the common difference and the 100th term of the following arithmetic sequence. 11, 6, 1, −4, −9, …

10. Ex 1. Find the common difference and the 100th term of the following arithmetic sequence. 11, 6, 1, −4, −9, …

11. Ex 1. Find the common difference and the 100th term of the following arithmetic sequence. 11, 6, 1, −4, −9, …

12. Ex 2. The 6th term of an arithmetic sequence is 10, and the 13th term is 31. Find the 𝑛th term and the 50th term.

13. Here’s a slick way to find the sum of the first 100 numbers:
𝑆 100 = …
𝑆 100 = …

14. Here’s a slick way to find the sum of the first 100 numbers:
𝑆 100 = …
𝑆 100 = …

15. Here’s a slick way to find the sum of the first 100 numbers:
𝑆 100 = …
𝑆 100 = …

16. Here’s a slick way to find the sum of the first 100 numbers:
𝑆 100 = …
𝑆 100 = …

17. Here’s a slick way to find the sum of the first 100 numbers:
𝑆 100 = …
𝑆 100 = …

18. Here’s a slick way to find the sum of the first 100 numbers:
𝑆 100 = …
𝑆 100 = …

19. Here’s a slick way to find the sum of the first 100 numbers:
𝑆 100 = …
𝑆 100 = …

20. Here’s a slick way to find the sum of the first 100 numbers:
𝑆 100 = …
𝑆 100 = …

21. Here’s a slick way to find the sum of the first 100 numbers:
𝑆 100 = …
𝑆 100 = …

22. Here’s a slick way to find the sum of the first 100 numbers:
𝑆 100 = …
𝑆 100 = …

23. This works for all arithmetic sequences, so that the sum of the first 𝑛 terms ( 𝑆 𝑛 ) is: 𝑺 𝒏 = 𝒂 𝟏 + 𝒂 𝒏 ⋅𝒏 𝟐 Ex 4. Find the sum: …+159

24. This works for all arithmetic sequences, so that the sum of the first 𝑛 terms ( 𝑆 𝑛 ) is: 𝑺 𝒏 = 𝒂 𝟏 + 𝒂 𝒏 ⋅𝒏 𝟐 Ex 3. Find the sum: …+159

25. This works for all arithmetic sequences, so that the sum of the first 𝑛 terms ( 𝑆 𝑛 ) is: 𝑺 𝒏 = 𝒂 𝟏 + 𝒂 𝒏 ⋅𝒏 𝟐 Ex 3. Find the sum: …+159

26. This works for all arithmetic sequences, so that the sum of the first 𝑛 terms ( 𝑆 𝑛 ) is: 𝑺 𝒏 = 𝒂 𝟏 + 𝒂 𝒏 ⋅𝒏 𝟐 Ex 3. Find the sum: …+159

27. This works for all arithmetic sequences, so that the sum of the first 𝑛 terms ( 𝑆 𝑛 ) is: 𝑺 𝒏 = 𝒂 𝟏 + 𝒂 𝒏 ⋅𝒏 𝟐 Ex 3. Find the sum: …+159

28. A geometric sequence is a sequence of the form: 𝑎, 𝑎𝑟, 𝑎 𝑟 2 , 𝑎 𝑟 3 , 𝑎 𝑟 4 , … 𝑎 is the first term, and 𝑟 is the common ratio. The 𝑛th term is given by 𝑎 𝑛 = _________

29. A geometric sequence is a sequence of the form: 𝑎, 𝑎𝑟, 𝑎 𝑟 2 , 𝑎 𝑟 3 , 𝑎 𝑟 4 , … 𝑎 is the first term, and 𝑟 is the common ratio. The 𝑛th term is given by 𝑎 𝑛 = _________

30. A geometric sequence is a sequence of the form: 𝑎, 𝑎𝑟, 𝑎 𝑟 2 , 𝑎 𝑟 3 , 𝑎 𝑟 4 , … 𝑎 is the first term, and 𝑟 is the common ratio. The 𝑛th term is given by 𝑎 𝑛 = _________

31. A geometric sequence is a sequence of the form: 𝑎, 𝑎𝑟, 𝑎 𝑟 2 , 𝑎 𝑟 3 , 𝑎 𝑟 4 , … 𝑎 is the first term, and 𝑟 is the common ratio. The 𝑛th term is given by 𝑎 𝑛 = _________
𝒂 𝒓 𝒏−𝟏

32. Ex 4. Find the common ratio and the nth term of the following geometric sequence. 2, −10, 50, −250, 1250, …

33. Ex 4. Find the common ratio and the nth term of the following geometric sequence. 2, −10, 50, −250, 1250, …

34. Ex 4. Find the common ratio and the nth term of the following geometric sequence. 2, −10, 50, −250, 1250, …

35. Ex 4. Find the common ratio and the nth term of the following geometric sequence. 2, −10, 50, −250, 1250, …

36. Ex 4. Find the common ratio and the nth term of the following geometric sequence. 2, −10, 50, −250, 1250, …

37. Subtract the equations to get: 𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛 𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
We can find the partial sums of a geometric sequence with another slick trick:
𝑆 𝑛 =𝑎+𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1
𝑟 𝑆 𝑛 = 𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1 +𝑎 𝑟 𝑛
Subtract the equations to get:
𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛
𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
𝑺 𝒏 = 𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓

38. Subtract the equations to get: 𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛 𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
We can find the partial sums of a geometric sequence with another slick trick:
𝑆 𝑛 =𝑎+𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1
𝑟 𝑆 𝑛 = 𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1 +𝑎 𝑟 𝑛
Subtract the equations to get:
𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛
𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
𝑺 𝒏 = 𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓

39. Subtract the equations to get: 𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛 𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
We can find the partial sums of a geometric sequence with another slick trick:
𝑆 𝑛 =𝑎+𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1
𝑟 𝑆 𝑛 = 𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1 +𝑎 𝑟 𝑛
Subtract the equations to get:
𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛
𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
𝑺 𝒏 = 𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓

40. Subtract the equations to get: 𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛 𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
We can find the partial sums of a geometric sequence with another slick trick:
𝑆 𝑛 =𝑎+𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1
𝑟 𝑆 𝑛 = 𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1 +𝑎 𝑟 𝑛
Subtract the equations to get:
𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛
𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
𝑺 𝒏 = 𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓

41. Subtract the equations to get: 𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛 𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
We can find the partial sums of a geometric sequence with another slick trick:
𝑆 𝑛 =𝑎+𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1
𝑟 𝑆 𝑛 = 𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1 +𝑎 𝑟 𝑛
Subtract the equations to get:
𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛
𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
𝑺 𝒏 = 𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓

42. Subtract the equations to get: 𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛 𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
We can find the partial sums of a geometric sequence with another slick trick:
𝑆 𝑛 =𝑎+𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1
𝑟 𝑆 𝑛 = 𝑎𝑟+𝑎 𝑟 2 +𝑎 𝑟 3 +𝑎 𝑟 4 +…+𝑎 𝑟 𝑛−1 +𝑎 𝑟 𝑛
Subtract the equations to get:
𝑆 𝑛 −𝑟 𝑆 𝑛 =𝑎−𝑎 𝑟 𝑛
𝑆 𝑛 1−𝑟 =𝑎(1− 𝑟 𝑛 )
𝑺 𝒏 = 𝒂 𝟏− 𝒓 𝒏 𝟏−𝒓

43. Ex 5. Find the sum: 4− − …−

44. Ex 5. Find the sum: 4− − …−

45. Ex 5. Find the sum: 4− − …−

46. Notice that the sum of the first 𝑛 terms of the previous example is 𝑆 𝑛 = 4⋅ 1− − 1 3 𝑛 1− − What happens as we add more and more (and more!) terms? In other words, what happens as 𝑛→∞?

47. Notice that the sum of the first 𝑛 terms of the previous example is 𝑆 𝑛 = 4⋅ 1− − 1 3 𝑛 1− − What happens as we add more and more (and more!) terms? In other words, what happens as 𝑛→∞?

48. Notice that the sum of the first 𝑛 terms of the previous example is 𝑆 𝑛 = 4⋅ 1− − 1 3 𝑛 1− − What happens as we add more and more (and more!) terms? In other words, what happens as 𝑛→∞?

49. Well, − 1 3 𝑛 →0, so 𝑆 𝑛 → 4⋅ 1−0 1− − 1 3 = So, the infinite number of terms add up to 3 4 ! 1− − … is an example of an infinite series.

50. Well, − 1 3 𝑛 →0, so 𝑆 𝑛 → 4⋅ 1−0 1− − 1 3 =3. So, the infinite number of terms add up to 3 4 ! 1− − … is an example of an infinite series.

51. Well, − 1 3 𝑛 →0, so 𝑆 𝑛 → 4⋅ 1−0 1− − 1 3 =3. So, the infinite number of terms add up to 3! 1− − … is an example of an infinite series.

52. Well, − 1 3 𝑛 →0, so 𝑆 𝑛 → 4⋅ 1−0 1− − 1 3 =3. So, the infinite number of terms add up to 3! 4− − … is an example of an infinite series.

53. To the right is a visual picture of the infinite geometric series 1 2 + 1 4 + 1 8 + 1 16 + 1 32 +…

54. In general, if −1<𝑟<1, then as 𝑛→∞, 𝑟 𝑛 →0, so 𝑆 𝑛 = 𝑎 1− 𝑟 𝑛 1−𝑟 → 𝑎 1−𝑟 . In this case, we say that the infinite series converges to 𝑎 1−𝑟 . However, if 𝑟≥1 or 𝑟≤−1, then the infinite series diverges.

55. In general, if −1<𝑟<1, then as 𝑛→∞, 𝑟 𝑛 →0, so 𝑆 𝑛 = 𝑎 1− 𝑟 𝑛 1−𝑟 → 𝑎 1−𝑟 . In this case, we say that the infinite series converges to 𝑎 1−𝑟 . However, if 𝑟≥1 or 𝑟≤−1, then the infinite series diverges.

56. Ex 6. Determine whether the infinite geometric series converges or diverges. If it converges, find its sum …

57. Ex 6. Determine whether the infinite geometric series converges or diverges. If it converges, find its sum …

58. Ex 6. Determine whether the infinite geometric series converges or diverges. If it converges, find its sum …

59. Ex 7. Determine whether the infinite geometric series converges or diverges. If it converges, find its sum …

60. Ex 7. Determine whether the infinite geometric series converges or diverges. If it converges, find its sum …

61. Ex 7. Determine whether the infinite geometric series converges or diverges. If it converges, find its sum …

62. Ex 7. Determine whether the infinite geometric series converges or diverges. If it converges, find its sum …