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Arithmetic Arithmetic Sequences Sequences

Arithmetic Sequences and Series

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Text of Arithmetic Sequences and Series

Page 1: Arithmetic Sequences and Series

Arithmetic Arithmetic SequencesSequences

Page 2: Arithmetic Sequences and Series
Page 3: Arithmetic Sequences and Series

An An Arithmetic SequenceArithmetic Sequence is is defineddefined as a sequence in as a sequence in which there is a which there is a common common

differencedifference between between consecutive terms.consecutive terms.

Page 4: Arithmetic Sequences and Series

Which of the following sequences are arithmetic? Identify the common

difference.

3, 1, 1, 3, 5, 7, 9, . . .

15.5, 14, 12.5, 11, 9.5, 8, . . .

84, 80, 74, 66, 56, 44, . . .

8, 6, 4, 2, 0, . . .

50, 44, 38, 32, 26, . . .

YES 2d

YES

YES

NO

NO

1.5d

6d

Page 5: Arithmetic Sequences and Series

The common

difference is

always the

difference between

any term and the

term that proceeds

that term.26, 21, 16, 11, 6, . . .

Common Difference = 5

Page 6: Arithmetic Sequences and Series

The general form of an ARITHMETIC sequence.

1aFirst Term:

Second Term: 2 1a a d

Third Term:

Fourth Term:

Fifth Term:

3 1 2a a d

4 1 3a a d

5 1 4a a d

nth Term: 1 1na a n d

Page 7: Arithmetic Sequences and Series

Formula for the nth term of an ARITHMETIC sequence.

1 1na a n d

The nth termna

The term numbern

The common differenced

1 The 1st terma

If we know any

If we know any three of these we

three of these we ought to be able

ought to be able to find the fourth.

to find the fourth.

Page 8: Arithmetic Sequences and Series

Given: 79, 75, 71, 67, 63, . . .Find: 32a

1 79

4

32

a

d

n

1

32

32

1

79 32 1 4

45

na a n d

a

a

IDENTIFY SOLVE

Page 9: Arithmetic Sequences and Series

Given: 79, 75, 71, 67, 63, . . .

Find: What term number is -169?

1 79

4

169n

a

d

a

1 1

169 79 1 4

63

na a n d

n

n

IDENTIFY SOLVE

Page 10: Arithmetic Sequences and Series

Given:10

12

3.25

4.25

a

a

1

3

3.25

4.25

3

a

a

n

1 1

4.25 3.25 3 1

0.5

na a n d

d

d

IDENTIFY SOLVE

Find: 1a

What’s the real question? The Difference

Page 11: Arithmetic Sequences and Series

Given:10

12

3.25

4.25

a

a

10 3.25

0.5

10

a

d

n

1

1

1

1

3.25 10 1 0.5

1.25

na a n d

a

a

IDENTIFY SOLVE

Find: 1a

Page 12: Arithmetic Sequences and Series

Arithmetic Arithmetic SeriesSeries

Page 13: Arithmetic Sequences and Series

Write the first three terms and the

Write the first three terms and the last two terms of the following

last two terms of the following arithmetic series.arithmetic series.

50

1

73 2p

p

71 69 67 . . . 25 27

What is the sum of What is the sum of

this series?this series?

Page 14: Arithmetic Sequences and Series

71 69 67 . . . 25 27

27 25 . . . 67 69 71

44 44 44 . . . 44 44 44

50 71 27

2

110071 + (-27) Each sum is the same.

50 Terms

Page 15: Arithmetic Sequences and Series

1 1 1 12 . . . 1a a d a d a n d

1 1 1 11 . . . 2a n d a d a d a

1

2nn a as

1

Sum

Number of Terms

First Term

Last Termn

S

n

a

a

1 1 1 1 1 11 1 . . . 1a a n d a a n d a a n d

Page 16: Arithmetic Sequences and Series

Find the sum of the terms of this arithmetic series.

35

1

29 3k

k

1

2nn a a

S

1

35

35

26

76

n

a

a

35 26 76

2875

S

S

Page 17: Arithmetic Sequences and Series

Find the sum of the terms of this arithmetic series. 151 147 143 139 . . . 5

1

2nn a a

S

1

40

40

151

5

n

a

a

40 151 5

22920

S

S

1 1

5 151 1 4

40

na a n d

n

n

What term is -5?What term is -5?

Page 18: Arithmetic Sequences and Series

Alternate formula for the

sum of an Arithmetic

Series.

1

2nn a

Sa

1 1Substitute na a n d

1 1

1

1

2

2 1

2

n a a n dS

n a n dS

1

# of Terms

1st Term

Difference

n

a

d

Page 19: Arithmetic Sequences and Series

Find the sum of this series 36

0

2.25 0.75j

j

2.25 3 3.73 4.5 . . .

12 1

2

n a n dS

It is not convenient to It is not convenient to find the last term.find the last term.

1

37

2.25

0.75

n

a

d

37 2 2.25 37 1 0.75

2582.75

S

S

Page 20: Arithmetic Sequences and Series

35

1

45 5i

i

1

2nn a a

S

12 1

2

n a n dS

135 40 130nn a a 135 40 5n a d

35 40 130

21575

S

S

35 2 40 35 1 3

21575

S

S