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# AA Section 7-5

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Geometric Sequences

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### AA Section 7-5

1. 1. Section 7-5 Geometric Sequences
2. 2. Warm-up Find the next three terms of each sequence. 2. 2, , , ,... 11 1 1. 14, 42, 126, 378, ... 28 32
3. 3. Warm-up Find the next three terms of each sequence. 2. 2, , , ,... 11 1 1. 14, 42, 126, 378, ... 28 32 1134, 3402, 10206
4. 4. Warm-up Find the next three terms of each sequence. 2. 2, , , ,... 11 1 1. 14, 42, 126, 378, ... 28 32 , , 1 1 1 1134, 3402, 10206 128 512 2048
5. 5. Examine the sequence... 48, 72, 108, 162, 243, 364.5, ... What pattern exists?
6. 6. Examine the sequence... 48, 72, 108, 162, 243, 364.5, ... What pattern exists? Multiplying by 1.5 between each term
7. 7. Geometric/Exponential Sequence:
8. 8. Geometric/Exponential Sequence: A sequence that has ﬁrst term g1 and a constant ratio r that you multiply by from term to term
9. 9. Geometric/Exponential Sequence: A sequence that has ﬁrst term g1 and a constant ratio r that you multiply by from term to term Constant Ratio:
10. 10. Geometric/Exponential Sequence: A sequence that has ﬁrst term g1 and a constant ratio r that you multiply by from term to term Constant Ratio: The number you multiply by from term to term; found by taking one term and dividing by the previous term; does not equal 0
11. 11. Recursive Formula for a Geometric Sequence  g1    gn = rgn−1, for int. n ≥ 2 
12. 12. Recursive Formula for a Geometric Sequence  g1    gn = rgn−1, for int. n ≥ 2  g1 = ﬁrst term
13. 13. Recursive Formula for a Geometric Sequence  g1    gn = rgn−1, for int. n ≥ 2  g1 = ﬁrst term gn = nth term
14. 14. Recursive Formula for a Geometric Sequence  g1    gn = rgn−1, for int. n ≥ 2  g1 = ﬁrst term gn = nth term gn -1 = previous term
15. 15. Recursive Formula for a Geometric Sequence  g1    gn = rgn−1, for int. n ≥ 2  g1 = ﬁrst term gn = nth term gn -1 = previous term r = constant ratio
16. 16. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2 
17. 17. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 =
18. 18. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 =
19. 19. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) =
20. 20. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12
21. 21. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 =
22. 22. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) =
23. 23. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36
24. 24. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 =
25. 25. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) =
26. 26. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) = 108
27. 27. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) = 108 g5 =
28. 28. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) = 108 g5 = 3(108) =
29. 29. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) = 108 g5 = 3(108) = 324
30. 30. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) = 108 g5 = 3(108) = 324 g6 =
31. 31. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) = 108 g5 = 3(108) = 324 g6 = 3(324) =
32. 32. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) = 108 g5 = 3(108) = 324 g6 = 3(324) = 972
33. 33. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) = 108 g5 = 3(108) = 324 g6 = 3(324) = 972 g7 =
34. 34. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) = 108 g5 = 3(108) = 324 g6 = 3(324) = 972 g7 = 3(972) =
35. 35. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 = 3(12) = 36 g4 = 3(36) = 108 g5 = 3(108) = 324 g6 = 3(324) = 972 g7 = 3(972) = 2916
36. 36. Explicit Formula for a Geometric Sequence n−1 , for int. n ≥ 1 gn = g1r
37. 37. Explicit Formula for a Geometric Sequence n−1 , for int. n ≥ 1 gn = g1r g1 = ﬁrst term
38. 38. Explicit Formula for a Geometric Sequence n−1 , for int. n ≥ 1 gn = g1r g1 = ﬁrst term gn = th term n
39. 39. Explicit Formula for a Geometric Sequence n−1 , for int. n ≥ 1 gn = g1r g1 = ﬁrst term gn = th term n r = constant ratio
40. 40. Question What is the difference between an arithmetic sequence and a geometric sequence?
41. 41. Question What is the difference between an arithmetic sequence and a geometric sequence? Arithmetic uses addition between terms; geometric uses multiplication.
42. 42. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1
43. 43. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1 1−1 g1 = 4(−2)
44. 44. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1 1−1 0 g1 = 4(−2) = 4(−2)
45. 45. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1 1−1 = 4(−2) = 4 0 g1 = 4(−2)
46. 46. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1 1−1 = 4(−2) = 4 0 g1 = 4(−2) 2−1 g2 = 4(−2)
47. 47. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1 1−1 = 4(−2) = 4 0 g1 = 4(−2) 2−1 1 g2 = 4(−2) = 4(−2)
48. 48. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1 1−1 = 4(−2) = 4 0 g1 = 4(−2) 2−1 1 g2 = 4(−2) = 4(−2) = −8
49. 49. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1 1−1 = 4(−2) = 4 0 g1 = 4(−2) 2−1 1 g2 = 4(−2) = 4(−2) = −8 3−1 2 g3 = 4(−2) = 4(−2) = 16
50. 50. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1 1−1 = 4(−2) = 4 0 g1 = 4(−2) 2−1 1 g2 = 4(−2) = 4(−2) = −8 3−1 2 g3 = 4(−2) = 4(−2) = 16 4−1 3 g4 = 4(−2) = 4(−2) = −32
51. 51. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1 1−1 = 4(−2) = 4 0 g1 = 4(−2) 2−1 1 g2 = 4(−2) = 4(−2) = −8 3−1 2 g3 = 4(−2) = 4(−2) = 16 4−1 3 g4 = 4(−2) = 4(−2) = −32 5−1 4 g5 = 4(−2) = 4(−2) = 64
52. 52. Example 3 A ball is dropped from a height of 6 m and bounces up to 85% of its previous height after each bounce. Let hn be the greatest height after the n th bounce. Find a formula for hn and the height after the 10th bounce.
53. 53. Example 3 A ball is dropped from a height of 6 m and bounces up to 85% of its previous height after each bounce. Let hn be the greatest height after the n th bounce. Find a formula for hn and the height after the 10th bounce. n−1 , for int. n ≥ 1 hn = h1r
54. 54. Example 3 A ball is dropped from a height of 6 m and bounces up to 85% of its previous height after each bounce. Let hn be the greatest height after the n th bounce. Find a formula for hn and the height after the 10th bounce. n−1 , for int. n ≥ 1 hn = h1r r = .85
55. 55. Example 3 A ball is dropped from a height of 6 m and bounces up to 85% of its previous height after each bounce. Let hn be the greatest height after the n th bounce. Find a formula for hn and the height after the 10th bounce. n−1 , for int. n ≥ 1 hn = h1r r = .85 h1 = height after ﬁrst bounce
56. 56. Example 3 A ball is dropped from a height of 6 m and bounces up to 85% of its previous height after each bounce. Let hn be the greatest height after the n th bounce. Find a formula for hn and the height after the 10th bounce. n−1 , for int. n ≥ 1 hn = h1r r = .85 h1 = height after ﬁrst bounce So what does 6 m represent?
57. 57. Example 3 A ball is dropped from a height of 6 m and bounces up to 85% of its previous height after each bounce. Let hn be the greatest height after the n th bounce. Find a formula for hn and the height after the 10th bounce. n−1 , for int. n ≥ 1 hn = h1r r = .85 h1 = height after ﬁrst bounce So what does 6 m represent? The initial height (h0)
58. 58. Example 3 A ball is dropped from a height of 6 m and bounces up to 85% of its previous height after each bounce. Let hn be the greatest height after the n th bounce. Find a formula for hn and the height after the 10th bounce. n−1 , for int. n ≥ 1 hn = h1r r = .85 h1 = height after ﬁrst bounce So what does 6 m represent? The initial height (h0) h1 = h0 (.85)
59. 59. Example 3 A ball is dropped from a height of 6 m and bounces up to 85% of its previous height after each bounce. Let hn be the greatest height after the n th bounce. Find a formula for hn and the height after the 10th bounce. n−1 , for int. n ≥ 1 hn = h1r r = .85 h1 = height after ﬁrst bounce So what does 6 m represent? The initial height (h0) h1 = h0 (.85) = 6(.85)
60. 60. Example 3 A ball is dropped from a height of 6 m and bounces up to 85% of its previous height after each bounce. Let hn be the greatest height after the n th bounce. Find a formula for hn and the height after the 10th bounce. n−1 , for int. n ≥ 1 hn = h1r r = .85 h1 = height after ﬁrst bounce So what does 6 m represent? The initial height (h0) h1 = h0 (.85) = 6(.85) = 5.1
61. 61. Example 3 n−1 , for int. n ≥ 1 hn = h1r
62. 62. Example 3 n−1 , for int. n ≥ 1 hn = h1r n−1 hn = 5.1(.85) , for int. n ≥ 1
63. 63. Example 3 n−1 , for int. n ≥ 1 hn = h1r n−1 hn = 5.1(.85) , for int. n ≥ 1 10−1 h10 = 5.1(.85)
64. 64. Example 3 n−1 , for int. n ≥ 1 hn = h1r n−1 hn = 5.1(.85) , for int. n ≥ 1 10−1 9 h10 = 5.1(.85) = 5.1(.85)
65. 65. Example 3 n−1 , for int. n ≥ 1 hn = h1r n−1 hn = 5.1(.85) , for int. n ≥ 1 10−1 9 = 5.1(.85) ≈ 1.18 m h10 = 5.1(.85)
66. 66. Homework
67. 67. Homework p. 447 #1-23 “It was on my ﬁfth birthday that Papa put his hand on my shoulder and said, ‘Remember, my son, if you ever need a helping hand, you’ll ﬁnd one at the end of your arm.’” - Sam Levenson