1. Arithmetic
and
Geometric
Sequences

2. Arithmetic
Sequences

3. Introduction

4. 3, 6, 9, 12, 15, 18, · · ·

5. 3, 6, 9, 12, 15, 18, · · ·
Called Terms

6.
3, 6, 9, 12, 15, 18, · · ·
We Number Them

7. And Use the Counting Number Symbol
3, 6, 9, 12, 15, 18, · · ·
And Use the Counting Number Symbol

8. And Use a Symbol for each term
3, 6, 9, 12, 15, 18, · · ·
And Use a Symbol for each term

9.
3, 6, 9, 12, 15, 18, · · ·
The General Symbol Is

10. Counting Number Symbol
General Term
3, 6, 9, 12, 15, 18, · · ·
Sequence
Symbols for each Term

11. Definition

12. Arithmetic Sequence There exists a constant d, such that
if and only if
There exists a constant d, such that
Where d is called the Common Difference

13. Arithmetic Sequence Recursive Formula
Where d is called the Common Difference

14. Example #1

15. How Do We Get From One Term To The Next?#1
1, 2, 3, 4, 5, 6, · · ·
How Do We Get From One Term To The Next?

16. #1 1 2 3 4 5 6 1, 2, 3, 4, 5, 6, · · · Arithmetic Sequence
1, 2, 3, 4, 5, 6, · · ·
Arithmetic Sequence
if and only if
There exists a constant d, such that

17. What is the Common Difference#1
1, 2, 3, 4, 5, 6, · · ·
Recursive Formula ?
What is the Common Difference

18. What is the Common Difference#1
1, 2, 3, 4, 5, 6, · · ·
Recursive Formula 1 ?
What is the Common Difference

19. #1
1, 2, 3, 4, 5, 6, · · ·
Recursive Formula 1

20. You need to know where the Sequence starts.
1, 2, 3, 4, 5, 6, · · · #1
Recursive Formula 1
You need to know where the Sequence starts.

21.
1, 2, 3, 4, 5, 6, · · · #1
Recursive Formula 1

22. Example #2

23. #2 1 2 3 4 5 6 2, 4, 6, 8, 10, 12, · · · Arithmetic Sequence
2, 4, 6, 8, 10, 12, · · ·
Arithmetic Sequence
if and only if
There exists a constant d, such that

24. #2
2, 4, 6, 8, 10, 12, · · ·
Recursive ?

25. #2
2, 4, 6, 8, 10, 12, · · ·
Recursive 2

26. #2
2, 4, 6, 8, 10, 12, · · ·
Recursive 2

27. Page 687 Class Exercises #1 to 8
Find which ones are Arithmetic Sequences and what is their common difference #1 #3 #6
d = 2
d = 13
d = -0.3

28. NEXT

29. For an Arithmetic Sequence
The Explicit Formula
For an Arithmetic Sequence
“machine”

36. The Explicit Formula ?

37. The Explicit Formula

38. Go Back to
Example #1

39.
1, 2, 3, 4, 5, 6, · · · #1
Recursive Formula

40. #1 1 2 3 4 5 6 1, 2, 3, 4, 5, 6, · · · Recursive Explicit Formula
1, 2, 3, 4, 5, 6, · · · #1
Common Difference is 1
Recursive
Explicit Formula

41. #1 1 2 3 4 5 6 1, 2, 3, 4, 5, 6, · · · Recursive Explicit Formula
1, 2, 3, 4, 5, 6, · · · #1
Common Difference is 1
Recursive
Explicit Formula

42. Now you can use the Explicit Formula to find any term in the Sequence.
1, 2, 3, 4, 5, 6, · · · #1
Explicit Formula
Now you can use the Explicit Formula to find any term in the Sequence.
For Example

43.
1, 2, 3, 4, 5, 6, · · · #1
Explicit Formula
33rd Term

44. Another
Example

45.
3, 7, 11, 15, 19, 23, · · ·
Common Difference is 4
Review First

46.
3, 7, 11, 15, 19, 23, · · ·
Common Difference is 4
Recursive
Review

47. 1 2 3 4 5 6 3, 7, 11, 15, 19, 23, · · · Recursive Explicit Formula
3, 7, 11, 15, 19, 23, · · ·
Common Difference is 4
Recursive
Explicit Formula

48. 1 2 3 4 5 6 3, 7, 11, 15, 19, 23, · · · Recursive Explicit Formula
3, 7, 11, 15, 19, 23, · · ·
Common Difference is 4
Recursive
Explicit Formula

49. Now you can use the Explicit Formula to find any term in the Sequence.
3, 7, 11, 15, 19, 23, · · ·
Common Difference is 4
Explicit Formula
Now you can use the Explicit Formula to find any term in the Sequence.
For Example

50.
3, 7, 11, 15, 19, 23, · · ·
Explicit Formula
101th Term

51. Geometric
Sequences

52. NO Common Difference 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · NOT
1, 2, 4, 8, 16, 32, · · ·
NO Common Difference
NOT
Arithmetic

53.
1, 2, 4, 8, 16, 32, · · ·
It May Be
Geometric

54. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

55. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

56. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

57. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

58. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

59. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

60. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

61. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

62. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

63. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

64. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

65. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

66. r is the Common Ratio 1 2 3 4 5 6 1, 2, 4, 8, 16, 32, · · · Geometric
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

67. r is the Common Ratio The Common Ratio is 2 1 2 3 4 5 6
1, 2, 4, 8, 16, 32, · · ·
Geometric
Test
r is the Common Ratio

68. Geometric
Test
r is the Common Ratio

69. Recursive
Geometric
Test
r is the Common Ratio

70. Recursive
Geometric
Test
r is the Common Ratio

71. Go Back

72.
1, 2, 4, 8, 16, 32, · · ·
Recursive
Recursive
The Common Ratio is 2

73.
1, 2, 4, 8, 16, 32, · · ·
Recursive

74.
1, 2, 4, 8, 16, 32, · · ·
Recursive

75. Definition

76. Geometric Sequence There exists a constant r, such that
if and only if
There exists a constant r, such that
Where r is called the Common Ratio

77. Explicit Formula ?

87. Explicit Term ?

88. Explicit Formula

89. Example

90.
2, 6, 18, 54, 162, 486, · · ·
Geometric Test

91.
2, 6, 18, 54, 162, 486, · · ·
Geometric Test

92.
2, 6, 18, 54, 162, 486, · · ·
Geometric Test

93.
2, 6, 18, 54, 162, 486, · · ·
Geometric Test
Common Ratio is 3

94.
2, 6, 18, 54, 162, 486, · · ·
Common Ratio is 3
Recursive

95. 1 2 3 4 5 6 2, 6, 18, 54, 162, 486, · · · Recursive Explicit Formula
2, 6, 18, 54, 162, 486, · · ·
Common Ratio is 3
Recursive
Explicit Formula

96. 1 2 3 4 5 6 2, 6, 18, 54, 162, 486, · · · Recursive Explicit Formula
2, 6, 18, 54, 162, 486, · · ·
Common Ratio is 3
Recursive
Explicit Formula

97. “machine” 1 2 3 4 5 6 2, 6, 18, 54, 162, 486, · · · Recursive
2, 6, 18, 54, 162, 486, · · ·
Common Ratio is 3
Recursive
Explicit Formula
“machine”

98. Next
Example

99.
32, -16, 8, -4, 2, -1, · · ·

100.
32, -16, 8, -4, 2, -1, · · ·

101.
32, -16, 8, -4, 2, -1, · · ·

102.
32, -16, 8, -4, 2, -1, · · ·

103.
32, -16, 8, -4, 2, -1, · · ·

104.
32, -16, 8, -4, 2, -1, · · ·

105.
32, -16, 8, -4, 2, -1, · · ·
Summary
Common Ratio is

106.
32, -16, 8, -4, 2, -1, · · ·
Recursive

107.
32, -16, 8, -4, 2, -1, · · ·
Recursive
Explicit Formula

108. Types of
Sequences

109. d = Common Difference r = Common Ratio NO Common Difference
Arithmetic
d = Common Difference
Geometric
r = Common Ratio
Neither
NO Common Difference
NO Common Ratio

110. Example

111.
2, 4, 5, 5, 4, 2, · · ·
NO Common Difference

112.
2, 4, 5, 5, 4, 2, · · ·

113.
2, 4, 5, 5, 4, 2, · · · vs

114.
2, 4, 5, 5, 4, 2, · · ·

115.
2, 4, 5, 5, 4, 2, · · ·

116.
2, 4, 5, 5, 4, 2, · · ·

117.
2, 4, 5, 5, 4, 2, · · ·
NO Common Ratio

118. 1 2 3 4 5 6 2, 4, 5, 5, 4, 2, · · · Neither NO Common Difference
2, 4, 5, 5, 4, 2, · · ·
NO Common Difference
NO Common Ratio
Neither

119. You Try

120. For The Following Problems
You Try
Determine if
Arithmetic
Geometric
Or Neither
For The Following Problems

121. Determine if Arithmetic Geometric or Neither

122. Determine if Arithmetic Geometric or Neither

123. Next

124. Looking at the Same Problems
You Try
Looking at the Same Problems
Find the common Difference or Common Ratio for
Arithmetic & Geometric
Only

125. Find the Common Difference or Common Ratio for Arithmetic & Geometric Only

126. Find the Common Difference or Common Ratio for Arithmetic & Geometric Only

127. Next

128. Looking at the Same Problems
You Try
Looking at the Same Problems
Find Explicit Formulas for
Arithmetic & Geometric
Only

129. Find Explicit Formulas for Arithmetic & Geometric Only

130. Find Explicit Formulas for Arithmetic & Geometric Only