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