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Arithmetic Arithmetic Sequences Sequences

# Arithmetic Sequences and Series

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Arithmetic Arithmetic SequencesSequences

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.

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

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

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

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.

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

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

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

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

Arithmetic Arithmetic SeriesSeries

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?

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

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

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

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?

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

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

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

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