Geometric Sequences and Series

1. Section 11-3
Geometric Sequences and Series

2. Essential Questions
• How do you use geometric
sequences?
• How do you find sums of geometric
series?

3. Vocabulary
1. Geometric Sequence:
2. Geometric Means:
3. Geometric Series:

4. Vocabulary
1. Geometric Sequence: A sequence where the next
term is found by multiplying the previous term by
a constant ratio r
2. Geometric Means:
3. Geometric Series:

5. Vocabulary
1. Geometric Sequence: A sequence where the next
term is found by multiplying the previous term by
a constant ratio r
an
= a1
rn−1
2. Geometric Means:
3. Geometric Series:

6. Vocabulary
1. Geometric Sequence: A sequence where the next
term is found by multiplying the previous term by
a constant ratio r
an
= a1
rn−1
2. Geometric Means: The terms between
nonconsecutive terms of a sequence
3. Geometric Series:

7. Vocabulary
1. Geometric Sequence: A sequence where the next
term is found by multiplying the previous term by
a constant ratio r
an
= a1
rn−1
2. Geometric Means: The terms between
nonconsecutive terms of a sequence
3. Geometric Series: The sum of the terms of a
geometric sequence

8. Vocabulary
4. Sum of a Geometric Series:

9. Vocabulary
4. Sum of a Geometric Series:
Sn
=
a1
− a1
rn
1− r
,r ≠ 1

10. Vocabulary
4. Sum of a Geometric Series:
Sn
=
a1
− a1
rn
1− r
,r ≠ 1
or

11. Vocabulary
4. Sum of a Geometric Series:
Sn
=
a1
− a1
rn
1− r
,r ≠ 1
Sn
=
a1
− an
r
1− r
,r ≠ 1
or

12. Example 1
Find the 6th term of the geometric sequence for the
following information.
a1
= −3, r = −2

13. Example 1
Find the 6th term of the geometric sequence for the
following information.
a1
= −3, r = −2
an
= a1
rn−1

14. Example 1
Find the 6th term of the geometric sequence for the
following information.
a1
= −3, r = −2
an
= a1
rn−1
a6
= (−3)(−2)6−1

15. Example 1
Find the 6th term of the geometric sequence for the
following information.
a1
= −3, r = −2
an
= a1
rn−1
a6
= (−3)(−2)6−1
a6
= (−3)(−2)5

16. Example 1
Find the 6th term of the geometric sequence for the
following information.
a1
= −3, r = −2
an
= a1
rn−1
a6
= (−3)(−2)6−1
a6
= (−3)(−2)5
a6
= (−3)(−32)

17. Example 1
Find the 6th term of the geometric sequence for the
following information.
a1
= −3, r = −2
an
= a1
rn−1
a6
= (−3)(−2)6−1
a6
= (−3)(−2)5
a6
= (−3)(−32)
a6
= 96

18. Example 2a
Write an equation for the nth term of each geometric
sequence.
5, 10, 20, 40,...

19. Example 2a
Write an equation for the nth term of each geometric
sequence.
5, 10, 20, 40,...
an
= a1
rn−1

20. Example 2a
Write an equation for the nth term of each geometric
sequence.
5, 10, 20, 40,...
an
= a1
rn−1
a1
= 5

21. Example 2a
Write an equation for the nth term of each geometric
sequence.
5, 10, 20, 40,...
an
= a1
rn−1
a1
= 5 r =
10
5
= 2

22. Example 2a
Write an equation for the nth term of each geometric
sequence.
5, 10, 20, 40,...
an
= a1
rn−1
a1
= 5 r =
10
5
= 2
an
= 5(2)n−1

23. Example 2b
Write an equation for the nth term of each geometric
sequence.
a5
= 4, r=3

24. Example 2b
Write an equation for the nth term of each geometric
sequence.
a5
= 4, r=3
an
= a1
rn−1

25. Example 2b
Write an equation for the nth term of each geometric
sequence.
a5
= 4, r=3
an
= a1
rn−1
4 = a1
(3)5−1

26. Example 2b
Write an equation for the nth term of each geometric
sequence.
a5
= 4, r=3
an
= a1
rn−1
4 = a1
(3)5−1
4 = a1
(3)4

27. Example 2b
Write an equation for the nth term of each geometric
sequence.
a5
= 4, r=3
an
= a1
rn−1
4 = a1
(3)5−1
4 = a1
(3)4
4 = a1
(81)

28. Example 2b
Write an equation for the nth term of each geometric
sequence.
a5
= 4, r=3
an
= a1
rn−1
4 = a1
(3)5−1
4 = a1
(3)4
4 = a1
(81)
4 = a1
(81)
8181

29. Example 2b
Write an equation for the nth term of each geometric
sequence.
a5
= 4, r=3
an
= a1
rn−1
4 = a1
(3)5−1
4 = a1
(3)4
4 = a1
(81)
4 = a1
(81)
8181
a1
=
4
81

30. Example 2b
Write an equation for the nth term of each geometric
sequence.
a5
= 4, r=3
an
= a1
rn−1
4 = a1
(3)5−1
4 = a1
(3)4
4 = a1
(81)
4 = a1
(81)
8181
a1
=
4
81
an
=
4
81
(3)n−1

31. Example 3
Find the three geometric means between 3.12 and
49.92.

32. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1

33. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12

34. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12 a5
= 49.92

35. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12 a5
= 49.92
49.92 = 3.12ir5−1

36. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12 a5
= 49.92
49.92 = 3.12ir5−1
3.123.12

37. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12 a5
= 49.92
49.92 = 3.12ir5−1
16 = r4
3.123.12

38. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12 a5
= 49.92
49.92 = 3.12ir5−1
16 = r4
3.123.12
± 16
4
= r44

39. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12 a5
= 49.92
49.92 = 3.12ir5−1
16 = r4
3.123.12
± 16
4
= r44
r = ±2

40. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12 a5
= 49.92
49.92 = 3.12ir5−1
16 = r4
3.123.12
± 16
4
= r44
r = ±2
a2
= 3.12(±2)

41. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12 a5
= 49.92
49.92 = 3.12ir5−1
16 = r4
3.123.12
± 16
4
= r44
r = ±2
a2
= 3.12(±2)
= ±6.24

42. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12 a5
= 49.92
49.92 = 3.12ir5−1
16 = r4
3.123.12
± 16
4
= r44
r = ±2
a2
= 3.12(±2)
= ±6.24
a3
= ±12.48

43. Example 3
Find the three geometric means between 3.12 and
49.92.
an
= a1
rn−1
a1
= 3.12 a5
= 49.92
49.92 = 3.12ir5−1
16 = r4
3.123.12
± 16
4
= r44
r = ±2
a2
= 3.12(±2)
= ±6.24
a3
= ±12.48
a4
= ±24.96

44. Example 4
Matt Mitarnowski hears a bad pun and retweets it,
tagging four of his followers. Each of those followers
retweets it and tag four more of their followers. This
pattern continues for a long time. It may have been a
bad pun, but it was punny! How many people are
tagged in the ninth round?

45. Example 4
Matt Mitarnowski hears a bad pun and retweets it,
tagging four of his followers. Each of those followers
retweets it and tag four more of their followers. This
pattern continues for a long time. It may have been a
bad pun, but it was punny! How many people are
tagged in the ninth round?
Sn
=
a1
− a1
rn
1− r

46. Example 4
Matt Mitarnowski hears a bad pun and retweets it,
tagging four of his followers. Each of those followers
retweets it and tag four more of their followers. This
pattern continues for a long time. It may have been a
bad pun, but it was punny! How many people are
tagged in the ninth round?
Sn
=
a1
− a1
rn
1− r
a1
= 4

47. Example 4
Matt Mitarnowski hears a bad pun and retweets it,
tagging four of his followers. Each of those followers
retweets it and tag four more of their followers. This
pattern continues for a long time. It may have been a
bad pun, but it was punny! How many people are
tagged in the ninth round?
Sn
=
a1
− a1
rn
1− r
a1
= 4 r = 4

48. Example 4
Matt Mitarnowski hears a bad pun and retweets it,
tagging four of his followers. Each of those followers
retweets it and tag four more of their followers. This
pattern continues for a long time. It may have been a
bad pun, but it was punny! How many people are
tagged in the ninth round?
Sn
=
a1
− a1
rn
1− r
a1
= 4 r = 4
n = 9

49. Example 4

50. Example 4
S9
=
4 − 4(4)9
1− 4

51. Example 4
S9
=
4 − 4(4)9
1− 4
S9
=
4 − 4(4)9
−3

52. Example 4
S9
=
4 − 4(4)9
1− 4
S9
=
4 − 4(4)9
−3
S9
=
4 − 4(262144)
−3

53. Example 4
S9
=
4 − 4(4)9
1− 4
S9
=
4 − 4(4)9
−3
S9
=
4 − 4(262144)
−3
S9
=
4 −1048576
−3

54. Example 4
S9
=
4 − 4(4)9
1− 4
S9
=
4 − 4(4)9
−3
S9
=
4 − 4(262144)
−3
S9
=
4 −1048576
−3
S9
=
−1048572
−3

55. Example 4
S9
=
4 − 4(4)9
1− 4
S9
=
4 − 4(4)9
−3
S9
=
4 − 4(262144)
−3
S9
=
4 −1048576
−3
S9
=
−1048572
−3
S9
= 349524

56. Example 4
S9
=
4 − 4(4)9
1− 4
S9
=
4 − 4(4)9
−3
S9
=
4 − 4(262144)
−3
S9
=
4 −1048576
−3
S9
=
−1048572
−3
S9
= 349524
retweets

57. Example 5
Find in a geometric series for the following.a1
S8
= 765 n = 8 r = 2

58. Example 5
Find in a geometric series for the following.a1
S8
= 765 n = 8 r = 2
Sn
=
a1
− a1
rn
1− r

59. Example 5
Find in a geometric series for the following.a1
S8
= 765 n = 8 r = 2
Sn
=
a1
− a1
rn
1− r
765 =
a1
− a1
(2)8
1−2

60. Example 5
Find in a geometric series for the following.a1
S8
= 765 n = 8 r = 2
Sn
=
a1
− a1
rn
1− r
765 =
a1
− a1
(2)8
1−2
765 =
a1
(1−28
)
−1

61. Example 5
Find in a geometric series for the following.a1
S8
= 765 n = 8 r = 2
Sn
=
a1
− a1
rn
1− r
765 =
a1
− a1
(2)8
1−2
765 =
a1
(1−28
)
−1
−765 = a1
(1−256)

62. Example 5
Find in a geometric series for the following.a1
S8
= 765 n = 8 r = 2
Sn
=
a1
− a1
rn
1− r
765 =
a1
− a1
(2)8
1−2
765 =
a1
(1−28
)
−1
−765 = a1
(1−256)
−765 = a1
(−255)

63. Example 5
Find in a geometric series for the following.a1
S8
= 765 n = 8 r = 2
Sn
=
a1
− a1
rn
1− r
765 =
a1
− a1
(2)8
1−2
765 =
a1
(1−28
)
−1
−765 = a1
(1−256)
−765 = a1
(−255)
a1
= 3

64. Example 6
3i2k−1
k=1
12
∑Find

65. Example 6
3i2k−1
k=1
12
∑Find
a1
= 3(2)1−1

66. Example 6
3i2k−1
k=1
12
∑Find
a1
= 3(2)1−1
= 3(2)0

67. Example 6
3i2k−1
k=1
12
∑Find
a1
= 3(2)1−1
= 3(2)0
= 3

68. Example 6
3i2k−1
k=1
12
∑Find
a1
= 3(2)1−1
= 3(2)0
= 3 n = 12−1+1 = 12

69. Example 6
3i2k−1
k=1
12
∑Find
a1
= 3(2)1−1
= 3(2)0
= 3 n = 12−1+1 = 12
Sn
=
a1
− a1
rn
1− r

70. Example 6
3i2k−1
k=1
12
∑Find
a1
= 3(2)1−1
= 3(2)0
= 3 n = 12−1+1 = 12
Sn
=
a1
− a1
rn
1− r
S12
=
3− 3(2)12
1−2

71. Example 6
3i2k−1
k=1
12
∑Find
a1
= 3(2)1−1
= 3(2)0
= 3 n = 12−1+1 = 12
Sn
=
a1
− a1
rn
1− r
S12
=
3− 3(2)12
1−2
S12
=
3− 3(4096)
−1

72. Example 6
3i2k−1
k=1
12
∑Find
a1
= 3(2)1−1
= 3(2)0
= 3 n = 12−1+1 = 12
Sn
=
a1
− a1
rn
1− r
S12
=
3− 3(2)12
1−2
S12
=
3− 3(4096)
−1
S12
=
3−12288
−1

73. Example 6

74. Example 6
S12
=
−12285
−1

75. Example 6
S12
=
−12285
−1
S12
= 12285