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# AA Section 7-5

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Geometric Sequences

Geometric Sequences

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• 1. Section 7-5 Geometric Sequences
• 2. Warm-up Find the next three terms of each sequence. 2. 2, , , ,... 11 1 1. 14, 42, 126, 378, ... 28 32
• 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. 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. Examine the sequence... 48, 72, 108, 162, 243, 364.5, ... What pattern exists?
• 6. Examine the sequence... 48, 72, 108, 162, 243, 364.5, ... What pattern exists? Multiplying by 1.5 between each term
• 7. Geometric/Exponential Sequence:
• 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. 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. 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. Recursive Formula for a Geometric Sequence  g1    gn = rgn−1, for int. n ≥ 2 
• 12. Recursive Formula for a Geometric Sequence  g1    gn = rgn−1, for int. n ≥ 2  g1 = ﬁrst term
• 13. Recursive Formula for a Geometric Sequence  g1    gn = rgn−1, for int. n ≥ 2  g1 = ﬁrst term gn = nth term
• 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. 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. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2 
• 17. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 =
• 18. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 =
• 19. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) =
• 20. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12
• 21. Example 1 Give the next 6 terms.  g1 = 4    gn = 3gn−1, for int. n ≥ 2  g2 = 3g1 = 3(4) = 12 g3 =
• 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Explicit Formula for a Geometric Sequence n−1 , for int. n ≥ 1 gn = g1r
• 37. Explicit Formula for a Geometric Sequence n−1 , for int. n ≥ 1 gn = g1r g1 = ﬁrst term
• 38. Explicit Formula for a Geometric Sequence n−1 , for int. n ≥ 1 gn = g1r g1 = ﬁrst term gn = th term n
• 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. Question What is the difference between an arithmetic sequence and a geometric sequence?
• 41. Question What is the difference between an arithmetic sequence and a geometric sequence? Arithmetic uses addition between terms; geometric uses multiplication.
• 42. Example 2 Write the ﬁrst 5 terms of the sequence n−1 gn = 4(−2) , for int. n ≥ 1
• 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Example 3 n−1 , for int. n ≥ 1 hn = h1r
• 62. Example 3 n−1 , for int. n ≥ 1 hn = h1r n−1 hn = 5.1(.85) , for int. n ≥ 1
• 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. 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. 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. Homework
• 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