1. Sequences
A2/trig

2. Sequences: Vocabulary
Sequence: an ordered list of numbers
Ex. -2, -1, 0, 1, 2, 3
Term: each number in a sequence
Ex. a1, a2, a3, a4, a5, a6
Recursive Formula: finding the next term depends on knowing a term or terms before it.
Explicit formula: defines the nth term of a sequence.

3. Vocabulary Recursive Formula: an-1
Uses one or more previous terms to generate the next term.
Any term: an
Previous term: an-1
an-1

4. Examples: A) Write the first six terms of the sequence
where a1 = -2 and an = 2an-1 – 1
( Always list the terms with subscripts first:)
a1, a2, a3, a4, a5, a6
B) Write the first six terms of the sequence where a1 = 4 and an = 3an-1 + 5

5. Examples: A) Write the first six terms of the sequence
where a1 = -2 and an = 2an-1 – 1
a1, a2, a3, a4, a5, a6
-2,-5,-11,-23,-47,-95
B) Write the first six terms of the sequence where a1 = 4 and an = 3an-1 + 5
a1, a2, a3, a4, a5, a6
4, 17, 56 ,173, 524, 1577

6. Recursive formula depending on two previous terms:
a1=-2, a2 = ak= ak-1 + ak-2
Find the first 6 terms.

7. Recursive formula depending on two previous terms:
a1=-2, a2 = ak= ak-1 + ak-2
Find the first 6 terms.
a1, a2, a3, a4, a5, a6
-2, 3, 1, 4, 5, 9

8. Some sequences are neither of these!
Two special sequence types: Arithmetic sequence:a sequence in which each term is found by adding a constant, called the common difference (d), to the previous term.
Geometric sequence: a sequence in which each term is founds by multiplying a constant, (r), called a common ratio to the previous term.
Some sequences are neither of these!

9. Do now: Recursive formula
Find the 10th term of a1 = 7 and an = an-1 + 6
Recursive formula

10. Example 1:
Find the 10th term of a1 = 7 and an = an-1 + 6
7,13,19,25,31,37,43,49,55,61

11. Formula for the nth term
Common difference
an = a1 + (n – 1)d
First term in the sequence
What term you are looking for
What term you are looking for

12. Example: Find the 10th term of a1 = 7 and an = an-1 + 6
Write the explicit formula
(recall a10 = 61)
an = a1 + d(n – 1)
an= 7+ 6(n-1)
an = 7 + 6n – 6
an = 6n + 1
a10 = 6(10) +1 = 61

13. Vocabulary Arithmetic Sequence: Start by asking, What is d? a2 – a1
A sequence generated by adding “d” a constant number to pervious term to obtain the next term.
This number is called the common difference.
Start by asking, What is d? a2 – a1
3, 7, 11, 15, … d = 4
8, 2, -4, -10, … d = -6

14. Find the explicit formula for these examples:
For Arithmetic Sequences, use the formula:
an = a1 + d(n – 1)
3, 7, 11, 15, … d = 4
8, 2, -4, -10, … d = -6

15. solutions: an = a1 + d(n – 1) an = 4n – 1 an = -6n + 14

16. Examples when a1 is not given
Find the 10th term of the arithmetic sequence where a3 = -5 and a6 = 16
Find the 15th term of the arithmetic sequence where a5 = 7 and a10 = 22
Find the 12th term of the arithmetic sequence where a3 = 8 and a7 = 20

17. Examples when a1 is not given
Find the 10th term of the arithmetic sequence where a3 = -5 and a6 = 16
= = 3
an = (n – 1)
an = n – 7
an = 7n – 26
A10 = 7(10)-26=44

18. Examples when a1 is not given
Find the 15th term of the arithmetic sequence where a5 = 7 and a10 = 22
22-7 = = 5
a1=-5 a2=-2 a3=1 a4=4 a5=7
an = (n – 1)
an = n – 3
an = 3n – 8
A15 = 3(15)-8=37

19. Examples when a1 is not given
Find the 12th term of the arithmetic sequence where a3 = 8 and a7 = 20
D = 3 a1 = 2
an = 2 + 3(n – 1)
an = 2 + 3n – 3
an = 3n – 1
A12 = 3(12)-1=35

20. Vocabulary Arithmetic Means: Terms in between 2 nonconsecutive terms
Ex. 5, 11, 17, 23, 29  11, 17, 23 are the arithmetic means between 5 & 29

21. Example 3: Find the 4 arithmetic means between 10 & -30

22. Example 3: Find the 4 arithmetic means between 10 & -30
10, 2, -6, -14,

23. Example 3: Find the 5 arithmetic means between 6 & 60
6, 15, 24, 33, 42, 51, 60
Example 3:

24. Geometric Sequences
multiplying

25. Do Now: Find the 5th term of a1 = 8 and an = 3an-1

26. Do Now: Find the 5th term of a1 = 8 and an = 3an-1 8, 24, 72, 216, 648
5, 10, 20, 40, 80, 160, 320

27. Vocabulary Geometric Sequence: What is r?
A sequence generated by multiplying a constant ratio to the previous term to obtain the next term.
This number is called the common ratio.
What is r?
2, 4, 8, 16, … r = 2
27, 9, 3, 1, … r = 1/3

28. Explicit Formula for the nth term
First term in the sequence
an = a1rn-1
What term you are looking for
What term you are looking for
Common Ratio

29. Explicit geometric formula an = a1rn-1
Find the Explicit formula and the 5th term of a1 = 8 and an = 3an-1
Find the Explicit formula and the 7th term of a1 = 5 and an = 2an-1

30. Explicit geometric formula
Find the explicit formula and the 5th term of a1 = 8 and an = 3an-1
an = a1rn-1
an = 8(3)n-1 a5 = 8(3)4 = 648
Find the Explicit formula and the 7th term of a1 = 5 and an = 2an-1
an = a1rn-1 a7 = 5(2)6 = 320

31. Warm up 1. Find the 8th term of the sequence defined by a1= –4 and
an= an-1+ 2
2. Find the 12th term of the arithmetic sequence in which
a4= 2 and a7= 6
3. Find the four arithmetic means between 6 and 26.
4. Find the 5th term on the sequence defined by a1= 2 and
an= 2an-1.

32. Summation

33. Series Series: the sum of a sequence Summation Notation:
EX. (for the above series)
End number
Formula to use
Start number

34. = 2(1) (2) (3) (4) -1 =
=16

35. Summation Properties
For sequences ak and bk and positive integer n:

36. Summation Formulas For all positive integers n: Constant Linear
Quadratic

37. Example 1:
Evaluate

38. Example 1:
Evaluate

39. Extra Example:
Evaluate

40. Extra Example:
Evaluate

41. Sum of an arithmetic sequence
Arithmetic Series
Sum of an arithmetic sequence

42. Do Now: add the terms of the 4 series above
Arithmetic Sequences
Geometric Sequences
ADD
To get next term
MULTIPLY
To get next term
Arithmetic Series
Sum of Terms
Geometric Series
Sum of Terms
Do Now: add the terms of the 4 series above

43. Do Now: add the terms of the 4 series above
Arithmetic Series
Sum of Terms
Geometric Series
Sum of Terms
Arithmetic Sequences
Geometric Sequences
ADD
To get next term
MULTIPLY
To get next term
Do Now: add the terms of the 4 series above

44. Formula for arithmetic series
Vocabulary
An Arithmetic Series is the sum of an arithmetic sequence.
Formula for arithmetic series
Sn=

45. Example 1: Find the series 1, 3, 5, 7, 9, 11
B. Find the series 8, 13, 18, 23, 28, 33, 38

46. Example 1: Find the series 1, 3, 5, 7, 9, 11
B. Find the series 8, 13, 18, 23, 28, 33, 38

47. Example 2: Given 3 + 12 + 21 + 30 + …, find S25
Find the 25th and the 11th terms by finding the explicit formula first.

48. Example 2: Given 3 + 12 + 21 + 30 + …, find S25
Now apply series formula..

49. Example 2: Given 3 + 12 + 21 + 30 + …, find S25

50. Example 3:
Evaluate

51. Formula for geometric series
Vocabulary
An Geometric Series is the sum of an geometric sequence.
Formula for geometric series
Sn=

52. Example 1:
Given the series …find S5

53. Example 1:
Given the series …find S5

54. Vocabulary of Sequences (Universal)

56. 1/2 7 x
127/128

57. Do Now:
Evaluate

58. Do Now:
Evaluate

59. Do Now: n
Evaluate r a1

60. Formula for a convergent infinite geometric series
Vocabulary
An Infinite Geometric Series is a geometric series with infinite terms.
Formula for a convergent infinite geometric series
S =
If r <1 then the _______ can be found (converges)
If r  1 then the _______ can’t be found (diverges)
SUM
SUM

61. Examples :
Find the sum of the infinite geometric series …
Find the partial sum (S4)
Determine the ratio
Find the sum of the infinite geometric series …

62. Example 1:
Find the sum of the infinite geometric series …
r = .4
B) Find the sum of the infinite geometric series …
r= 1.2 so it is divergent

63. Example 2:
Find the sum of the infinite geometric series below:

64. Example 2:
Find the sum of the infinite geometric series below:
r =

65. Example 3: Write 0.2 as a fraction in simplest form.

66. Example 3: Write 0.2 as a fraction in simplest form.