Arithmetic Arithmetic Sequences Sequences and Seriesand Series

Sequences Series

List with commas “Indicated sum”

3, 8, 13, 18 3 + 8 + 13 + 18

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

An introduction…………

1, 4, 7,10,13

9,1, 7, 15

6.2, 6.6, 7, 7.4

, 3, 6

Arithmetic Sequences

ADDTo get next term

2, 4, 8,16, 32

9, 3,1, 1/ 3

1,1/ 4,1/16,1/ 64

, 2.5 , 6.25

Geometric Sequences

MULTIPLYTo get next term

Arithmetic Series

Sum of Terms

35

12

27.2

3 9

Geometric Series

Sum of Terms

62

20 / 3

85 / 64

9.75

Find the next four terms of –9, -2, 5, …

Arithmetic Sequence

2 9 5 2 7

7 is referred to as the common difference (d)

Common Difference (d) – what we ADD to get next term

Next four terms……12, 19, 26, 33

Find the next four terms of 0, 7, 14, …

Arithmetic Sequence, d = 7

21, 28, 35, 42

Find the next four terms of x, 2x, 3x, …

Arithmetic Sequence, d = x

4x, 5x, 6x, 7x

Find the next four terms of 5k, -k, -7k, …

Arithmetic Sequence, d = -6k

-13k, -19k, -25k, -32k

Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

n 1

n 1 n

nth term of arithmetic sequence

sum of n terms of arithmetic sequen

a a n 1 d

nS a a

2ce

Given an arithmetic sequence with 15 1a 38 and d 3, find a .

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

x

15

38

NA

-3

n 1a a n 1 d

38 x 1 15 3

X = 80

63Find S of 19, 13, 7,...

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

-19

63

??

x

6

n 1a a n 1 d

?? 19 6 1

?? 353

3 6

353

n 1 n

nS a a

2

63

633 3S

219 5

63 1 1S 052

16 1Find a if a 1.5 and d 0.5 Try this one:

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

1.5

16

x

NA

0.5

n 1a a n 1 d

16 1.5 0.a 16 51

16a 9

n 1Find n if a 633, a 9, and d 24

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

9

x

633

NA

24

n 1a a n 1 d

633 9 21x 4

633 9 2 244x

X = 27

1 29Find d if a 6 and a 20

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

-6

29

20

NA

x

n 1a a n 1 d

120 6 29 x

26 28x

13x

14

Find two arithmetic means between –4 and 5

-4, ____, ____, 5

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

-4

4

5

NA

x

n 1a a n 1 d

15 4 4 x x 3

The two arithmetic means are –1 and 2, since –4, -1, 2, 5

forms an arithmetic sequence

Find three arithmetic means between 1 and 4

1, ____, ____, ____, 4

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

1

5

4

NA

x

n 1a a n 1 d

4 1 x15 3

x4

The three arithmetic means are 7/4, 10/4, and 13/4

since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence

Find n for the series in which 1 na 5, d 3, S 440

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

5

x

y

440

3

n 1a a n 1 d

n 1 n

nS a a

2

y 5 31x

x40 y4

25

12

x440 5 5 x 3

x 7 x440

2

3

880 x 7 3x 20 3x 7x 880

X = 16

Graph on positive window

Example: The nth Partial Sum

The sum of the first n terms of an infinite sequence is called the nth partial sum.

1( )2n nnS a a

Example 6. Find the 150th partial sum of the arithmetic sequence, 5, 16, 27, 38, 49, …

1 5 11 5 11 6a d c

11 6na n 150 11 150 6 1644a

150

1505 1644 75 1649 123,675

2S

Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows?

1 20 1 19d c

1 201 20 19 1 39na a n d a

20

2020 39 10 59 590

2S

Example 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation.

1 10,000 7500 10,000 7500 2500a d c

1 201 10,000 19 7500 152,500na a n d a

20

2010,000 152,500 10 162,500 1,625,000

2S

So the total sales for the first 2o years is $1,625,000