1. Series NOTES
Name ____________________________
Arithmetic
Sequences

2. USING AND WRITING SEQUENCES
The numbers in sequences are called terms.
You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.

3. USING AND WRITING SEQUENCES
DOMAIN:
The domain gives the relative position of each term.
The range gives the terms of the sequence.
RANGE:
This is a finite sequence having the rule
an = 3n,
where an represents the nth term of the sequence.

4. Write the first six terms of the sequence an = 2n + 3.
Writing Terms of Sequences
Write the first six terms of the sequence an = 2n + 3.
SOLUTION
a 1 = 2(1) + 3 = 5
1st term
a 2 = 2(2) + 3 = 7
2nd term
a 3 = 2(3) + 3 = 9
3rd term
a 4 = 2(4) + 3 = 11
4th term
a 5 = 2(5) + 3 = 13
5th term
a 6 = 2(6) + 3 = 15
6th term

5. f (1) = (–2) 1 – 1 = 1 f (2) = (–2) 2 – 1 = –2 f (3) = (–2) 3 – 1 = 4
Writing Terms of Sequences
Write the first six terms of the sequence f (n) = (–2) n – 1 .
SOLUTION
f (1) = (–2) 1 – 1 = 1
1st term
f (2) = (–2) 2 – 1 = –2
2nd term
f (3) = (–2) 3 – 1 = 4
3rd term
f (4) = (–2) 4 – 1 = – 8
4th term
f (5) = (–2) 5 – 1 = 16
5th term
f (6) = (–2) 6 – 1 = – 32
6th term

6. d = 3 r =2 r = d = -8 r = d = .4 d = r = ARITHMETIC ADD
An introduction…………
d = 3
r =2
r =
d = -8
r =
d = .4
d =
r =
ARITHMETIC
ADD
(by the same #)
To get the next term
GEOMETRIC
MULTIPLY
(by the same #)
To get the next term

7. Finite VS. Infinite an-1 previous term an+1 next term
Vocabulary of Sequences (Universal)
an-1 previous term an+1 next term
Finite VS. Infinite

8. The terms have a common difference of 2. (known as d)
Arithmetic Sequence: sequence whose consecutive terms have a common difference.
Example: 3, 5, 7, 9, 11, 13, ...
The terms have a common difference of (known as d)
To find the common difference you use an+1 – an
Example: Is the sequence arithmetic? If so, find d. –45, –30, –15, 0, 15, 30
d = 15

9. Next four terms…… 12, 19, 26, 33 Find the next 4 terms of –9, -2, 5, …
7 is referred to as d
Next four terms……
12, 19, 26, 33

10. Find the next four terms of 0, 7, 14, …
Find the next four terms of x, 2x, 3x, …
Find the next four terms of 5k, -k, -7k, …
Arithmetic Sequence, d = 7
21, 28, 35, 42
Arithmetic Sequence, d = x
4x, 5x, 6x, 7x
Arithmetic Sequence, d = -6k
-13k, -19k, -25k, -31k

11. 4, 10, 16, 22
The nth term of an arithmetic sequence is given by:
The nth term in the sequence
The common difference
The term #
First term
Find the 10th term:

12. Find the 14th term of
the sequence:
Examples:
4, 7, 10, 13,……

13. Examples:
In the arithmetic sequence 4,7,10,13,…, which term has a value of 301?

14. Given an arithmetic sequence with
X = 80

16. Example: If the common difference is 4 and the fifth term is 15, what is the 10th term of an arithmetic sequence?
an = a1 + (n – 1)d
d = 4, a5 = 15, n = 5, a1=?
15 = a1 + (5 – 1)4
15 = a1 +16
a1 = –1
a10 = –1 + (10 – 1)4 =
a10 = 35

17. Ex: 4, 6, 8, 10… Explicit vs. Recursive Formulas
Explicit Formula – used to find the nth term of the arithmetic sequence in which the common difference and 1st term are known.
Ex: 4, 6, 8, 10…
Use a1 and d in sequence formula:
an = 4 + (n – 1)2 an = 2n + 2

18. Find the explicit formula for the following arithmetic sequence:
3, 8, 13, 18…
an = a1 + (n – 1)d a1 = 3 d = 5 n = ?
an = 3 + (n – 1)5
an = 3 + 5n – 5
an = n OR an = 5n – 2

19. an = an-1 + 2 an = an-1 + d a1 = ___ an+1 = an + d a1 = 4
Explicit vs. Recursive Formulas
Recursive Formula – (includes a1) used to find the next term of the sequence by adding the common difference to the previous term.
an = an-1 + d
a1 = ___
an+1 = an + d
an = an-1 + 2
a1 = 4
Ex: 4, 6, 8, 10…

20. a1 = 3 an = an-1 + d a1 = 3 d = 5 an = an-1 + 5
Series NOTES
Name ____________________________
Find the recursive formula for the following arithmetic sequence:
3, 8, 13, 18…
an = an-1 + d a1 = 3 d = 5
an = an-1 + 5
a1 = 3

21. an = an-1 + 6 a1 = 4 Using Recursive & Explicit Formulas
1. Create the 1st 5 terms:
4, 10, 16, 22, 28
a2 = = 10
2. Find the explicit formula:
an = a1 + (n – 1)d
a3 = = 16
an = 4 + (n – 1)6
a4 = = 22
an = 4 + 6n – 6
an = 6n – 2
a5 = = 28

22. Using Recursive & Explicit Formulas
an = 7 – 2n
1. Create the 1st 5 terms:
5, 3, 1, –1, –3
a1 = 7 – 2(1) = 5
a2 = 7 – 2(2) = 3
2. Find the
recursive formula:
a3 = 7 – 2(3) = 1
a4 = 7 – 2(4) = –1
an = an-1 – 2
a1 = 5
a5 = 7 – 2(5) = –3

23. Examples: Insert 3 arithmetic means between 8 & 16.
An arithmetic mean of two numbers, a and b, is simply their average. Use the formula and information given to find the common difference to create the sequence.
Examples:
Insert 3 arithmetic means between 8 & 16. 14 10 12
Let 8 be the 1st term
Let 16 be the 5th term
Let 5 be N
d is missing

24. The two arithmetic means are –1 and 2,
Find two arithmetic means
between –4 and 5
-4, ____, ____, 5
The two arithmetic means are –1 and 2,
since –4, -1, 2, 5
forms an arithmetic sequence

25. Find 3 arithmetic means between 1 & 4
1, ____, ____, ____, 4
The 3 arithmetic means are
since 1, ,4 forms a sequence

26. Geometric
Sequences

27. Finite VS. Infinite an-1 previous term an+1 next term
Series NOTES
Name ____________________________
Vocabulary of Sequences (Universal)
an-1 previous term an+1 next term
Finite VS. Infinite

28. Use to determine common ratio
Find the next 3 terms of 2, 3, 9/2, __, __, __
3 – 2 vs. 9/2 – 3… not arithmetic
Use to determine common ratio

29. 1st term: 2 4th term: 5th term: 6th term: How is the formula derived?
The nth term of a
geometric sequence is given by:
1st term: 2
4th term:
5th term:
6th term:

31. -3, ____, ____, ____

32. r =
a1=
n = 9

33. )2-1 ( 2 8 - = a x =

34. Ex: 4, 12, 36, 108… Use a1 and r in sequence formula:
Explicit vs. Recursive Formulas
Explicit Formula – used to find the nth term of the geometric sequence in which the common ratio and 1st term are known.
Ex: 4, 12, 36, 108…
Use a1 and r in sequence formula:
Ex: an = a1*rn an = 4 * 3n-1

35. Find the explicit formula for the following geometric sequence:
3, 6, 12, 24…
an = a1*rn a1 = 3 r =2
an = 3 *2n-1

36. an+1 = r(an) an = an-1 (r) a1 = ___ an = an-1 (–4) a1 = –1 a1 (r) = a2
Explicit vs. Recursive Formulas
Recursive Formula (includes a1) – used to find the next term of the sequence by multiplying the common ratio to the previous term.
an = an-1 (r)
a1 = ___
an+1 = r(an)
Ex: –1, 4, –16, 64 …
a1 (r) = a2
an = an-1 (–4)
a1 = –1
a2 (r) = a3
a3 (r) = a4

37. a1 = 3 an = an-1 * r a1 = 3 r = 2 an = an-1 * 2
Series NOTES
Name ____________________________
Find the recursive formula for the following geometric sequence:
3, 6, 12, 24…
an = an-1 * r a1 = 3 r = 2
an = an-1 * 2
a1 = 3

38. an = an-1 (3) a1 = –1 Using Recursive & Explicit Formulas
1. Create the 1st 5 terms:
–1, –3, –9, –27, – 81
a2 = –1(3) = –3
2. Find the explicit formula:
an = a1 (r)n-1
a3 = –3(3) = –9
an = –1(3)n-1
a4 = –9(3) = –27
an = –3n-1
a5 = –27(3) = –81

39. Using Recursive & Explicit Formulas
an = 2(4)n – 1
1. Create the 1st 5 terms:
2, 8, 32, 128, 512
a1 = 2(4)1-1 = 2
a2 = 2(4)2-1 = 8
2. Find the
recursive formula:
a3 = 2(4)3-1 = 32
a4 = 2(4)4-1 = 128
an = 4an-1
a1 = 2
a5 = 2(4)5-1 = 512

40. 6 –18 Ex: Find two geometric means between –2 and 54
A geometric mean(s) of numbers are the terms between any 2 nonsuccessive terms of a geometric sequence. Use the terms given to find the common ratio and find the missing terms called the geometric means.
Ex: Find two
geometric means
between –2 and 54
The 2 geometric means
are 6 and -18 6
–18
-2, ____, ____, 54

41. *** Insert one geometric mean between ¼ and 4***
Series NOTES
Name ____________________________
*** Insert one geometric mean between ¼ and 4***
*** denotes trick question

42. Series

43. Finite VS. Infinite an-1 previous term an+1 next term
Series NOTES
Name ____________________________
Vocabulary of Sequences (Universal)
an-1 previous term an+1 next term
Finite VS. Infinite

44. USING SERIES
When the terms of a sequence are added, the resulting expression is a series. A series can be finite or infinite.
FINITE SEQUENCE
FINITE SERIES
3, 6, 9, 12, 15
INFINITE SEQUENCE
INFINITE SERIES
3, 6, 9, 12, 15, . . .
. . .
You can use summation notation to write a series. For example, for the finite series shown above, you can write
= ∑ 3i 5
i = 1

45. # of Terms: B – A + 1 UPPER BOUND TERM NUMBER SIGMA NTH TERM
(SUM OF TERMS)
NTH TERM
SEQUENCE
(EXPLICIT FORMULA)
LOWER BOUND
TERM NUMBER

47. It can be infinite or finite.
An arithmetic series is a series associated with an arithmetic sequence.
It can be infinite or finite.
Definition:

48. 1, 4, 7, 10, 13, …. No Sum 3, 7, 11, …, 51 Infinite Arithmetic
(constantly getting
larger or smaller)
1, 4, 7, 10, 13, ….
No Sum
3, 7, 11, …, 51
Finite Arithmetic
1, 2, 4, …, 64
1, 2, 4, 8, …

49. Find the sum of the 1st Examples: 100 natural numbers.
… + 100

50. S14 = a14 = 2 + (14 - 1)(3) = 41 Find the sum of the 1st Examples:
14 terms of the series: …
To find a14 , you need
a14 = 2 + (14 - 1)(3) = 41
S14 =

51. Examples:
Find the sum of the series
Need 13th term:
4(13) + 5 = 57

52. Finding the Sum from Summation Notation
n = a1 = a4 = 6
3, 4, 5, 6
n = (7 – 4) a4 = a7 = 14
8, 10, 12, 14

53. 19, 23, 27, 31…79
a4 = a19 = n = (19 - 4) + 1 = 16
15, 17, 19, …47
a7 =15 a23 = n = (23-7) + 1 = 17

54. An geometric series is a series associated with a geometric sequence.
They can be infinite or finite.
Finite and infinite have different formulas depending on the value of r.
Definition:

55. No Sum No Sum 1, 2, 4, …, 64 1, 2, 4, 8, … Finite Geometric
1, 4, 7, 10, 13, ….
Infinite Arithmetic
(constantly getting
larger or smaller)
No Sum
3, 7, 11, …, 51
Finite Arithmetic
1, 2, 4, …, 64
Finite Geometric
Infinite Geometric
r < -1 OR r > 1
(constantly getting
larger or smaller)
“diverges”
1, 2, 4, 8, …
No Sum
Infinite Geometric
-1 < r < 1
“converges”

57. Sums of Infinite Series Made Finite (referred to as partial sums)
Finding the Sum
of Infinite Sequences
“Converges” vs. “Diverges”

59. Find the sum, if possible:
Geometric
~need to find r~
Is -1 < r < 1? Yes (Infinite Series - converges)

60. Find the sum, if possible:
Is -1 < r < 1? No (Infinite series - Diverges)

61. Find the sum, if possible:
Is -1 < r < 1? Yes (Infinite Series – Converges)

62. Find the sum, if possible:
Is -1 < r < 1? No (Infinite Series–Diverges)

63. Find the sum, if possible:
-1 < r < Yes (Infinite Series–Converges)

64. Finding the Sum from Sigma Notation
so “converges”

65. 4 n=1 1st term 4th term Arithmetic, d= 3
Rewrite using sigma notation:
1st term
n=1
4th term 4
Arithmetic, d= 3
Explicit formula

66. Geometric, r = ½ 5 n=1 1st term 5th term Explicit formula
Rewrite using sigma notation:
n=1
1st term 5
5th term
Geometric, r = ½
Explicit formula

67. SIGMA NOTATION: NUMERATOR: DENOMINATOR:
Rewrite the following using sigma notation:
Numerator is geometric, r = 3
Denominator is arithmetic d= 5
NUMERATOR:
DENOMINATOR:
SIGMA NOTATION: