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This ppt. provides an introduction to Arithmetic Sequence and Series

1.
Arithmetic Sequences and Series

2.
An introduction…………

1, 4, 7, 10, 13 35 2, 4, 8, 16, 32 62

9, 1, 7, 15 12 9, 3, 1, 1/ 3 20 / 3

6.2, 6.6, 7, 7.4 27.2 1, 1/ 4, 1/16, 1/ 64 85 / 64

, 3, 6 3 9 , 2.5, 6.25 9.75

Arithmetic Sequences Geometric Sequences

ADD MULTIPLY

To get next term To get next term

Arithmetic Series Geometric Series

Sum of Terms Sum of Terms

1, 4, 7, 10, 13 35 2, 4, 8, 16, 32 62

9, 1, 7, 15 12 9, 3, 1, 1/ 3 20 / 3

6.2, 6.6, 7, 7.4 27.2 1, 1/ 4, 1/16, 1/ 64 85 / 64

, 3, 6 3 9 , 2.5, 6.25 9.75

Arithmetic Sequences Geometric Sequences

ADD MULTIPLY

To get next term To get next term

Arithmetic Series Geometric Series

Sum of Terms Sum of Terms

3.
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.

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.

4.
USING AND WRITING SEQUENCES

n

DOMAIN: 1 2 3 4 5 The domain gives

the relative position

of each term.

The range gives the

an

RANGE: 3 6 9 12 15

terms of the sequence.

This is a finite sequence having the rule

an = 3n,

where an represents the nth term of the sequence.

n

DOMAIN: 1 2 3 4 5 The domain gives

the relative position

of each term.

The range gives the

an

RANGE: 3 6 9 12 15

terms of the sequence.

This is a finite sequence having the rule

an = 3n,

where an represents the nth term of the sequence.

5.
Writing Terms of Sequences

Write the first five terms of the sequence an = 2n + 3.

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

Write the first five terms of the sequence an = 2n + 3.

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

6.
Writing Terms of Sequences

n

Write the first five terms of the sequence an n 1

.

(1) 1

a1 1st term

(1) 1 2

( 2) 2

a2 2nd term

(2) 1 3

(3) 3

a3 3rd term

(3) 1 4

(4) 4

a4 4th term

(4) 1 5

(5) 5

a5 5th term

(5) 1 6

n

Write the first five terms of the sequence an n 1

.

(1) 1

a1 1st term

(1) 1 2

( 2) 2

a2 2nd term

(2) 1 3

(3) 3

a3 3rd term

(3) 1 4

(4) 4

a4 4th term

(4) 1 5

(5) 5

a5 5th term

(5) 1 6

7.
Writing Terms of Sequences

Write the first five terms of the sequence an = n! - 2.

a 1 = (1)!-2 = -1 1st term

a 2 = (2)! - 2 = 0 2nd term

a 3 = (3)! - 2 = 4 3rd term

a 4 = (4)! - 2 = 22 4th term

a 5 = (5)! - 2 = 118 5th term

Write the first five terms of the sequence an = n! - 2.

a 1 = (1)!-2 = -1 1st term

a 2 = (2)! - 2 = 0 2nd term

a 3 = (3)! - 2 = 4 3rd term

a 4 = (4)! - 2 = 22 4th term

a 5 = (5)! - 2 = 118 5th term

8.
Writing Terms of Sequences

Write the first five terms of the sequence f (n) = (–2) n – 1 .

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

Write the first five terms of the sequence f (n) = (–2) n – 1 .

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

9.
Arithmetic Sequences and Series

Arithmetic Sequence: sequence whose

consecutive terms have a common

difference.

Example: 3, 5, 7, 9, 11, 13, ...

The terms have a common difference of 2.

The common difference is the number d.

To find the common difference you use an+1

– an

Example: Is the sequence arithmetic?

–45, –30, –15, 0, 15, 30

Yes, the common difference is 15

Arithmetic Sequence: sequence whose

consecutive terms have a common

difference.

Example: 3, 5, 7, 9, 11, 13, ...

The terms have a common difference of 2.

The common difference is the number d.

To find the common difference you use an+1

– an

Example: Is the sequence arithmetic?

–45, –30, –15, 0, 15, 30

Yes, the common difference is 15

10.
Find the next four terms of –9, -2, 5, …

Arithmetic Sequence

-2 - -9 = 7 and 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

Arithmetic Sequence

-2 - -9 = 7 and 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

11.
Find the next four terms of 0, 7, 14, …

Arithmetic Sequence, d = 7

21, 28, 35, 42

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

Arithmetic Sequence, d = x

0, x, 2x, 3x

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

Arithmetic Sequence, d = -6k

-13k, -19k, -25k, -31k

Arithmetic Sequence, d = 7

21, 28, 35, 42

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

Arithmetic Sequence, d = x

0, x, 2x, 3x

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

Arithmetic Sequence, d = -6k

-13k, -19k, -25k, -31k

12.
How do you find any term in this sequence?

To find any term in an arithmetic sequence, use the formula

an = a1 + (n – 1)d

where d is the common difference.

To find any term in an arithmetic sequence, use the formula

an = a1 + (n – 1)d

where d is the common difference.

13.
Vocabulary of Sequences (Universal)

a1 First term

an nth term

n number of terms

Sn sum of n terms

d common difference

nth term of arithmetic sequence an a1 n 1 d

n

sum of n terms of arithmetic sequence Sn a1 an

2

a1 First term

an nth term

n number of terms

Sn sum of n terms

d common difference

nth term of arithmetic sequence an a1 n 1 d

n

sum of n terms of arithmetic sequence Sn a1 an

2

14.
Find the 14th term of the

arithmetic sequence

4, 7, 10, 13,……

an a1 (n 1)d

a14 4 (14 1)3

4 (13)3

4 39

43

arithmetic sequence

4, 7, 10, 13,……

an a1 (n 1)d

a14 4 (14 1)3

4 (13)3

4 39

43

15.
Try this one: Find a16 if a1 1.5 and d 0.5

1.5 a1 First term

a16 an nth term

16 n number of terms

NA Sn sum of n terms

0.5 d common difference

an a1 n 1 d

a16 = 1.5 + (16 - 1)(0.5)

a16 = 1.5 + (15)(0.5)

a16 = 1.5+7.5

a16 = 9

1.5 a1 First term

a16 an nth term

16 n number of terms

NA Sn sum of n terms

0.5 d common difference

an a1 n 1 d

a16 = 1.5 + (16 - 1)(0.5)

a16 = 1.5 + (15)(0.5)

a16 = 1.5+7.5

a16 = 9

16.
The table shows typical costs for a construction company to rent a

crane for one, two, three, or four months. Assuming that the arithmetic

sequence continues, how much would it cost to rent the crane for

twelve months?

Months Cost ($) 75,000 a1 First term an a1 n 1 d

1 75,000 a12 an nth term a12 = 75,000 + (12 - 1)(15,000)

2 90,000

3 105,000 12 n number of terms a12 = 75,000 + (11)(15,000)

4 120,000 NA Sn sum of n terms

a12 = 75,000+165,000

15,000 d common difference

a12 = $240,000

crane for one, two, three, or four months. Assuming that the arithmetic

sequence continues, how much would it cost to rent the crane for

twelve months?

Months Cost ($) 75,000 a1 First term an a1 n 1 d

1 75,000 a12 an nth term a12 = 75,000 + (12 - 1)(15,000)

2 90,000

3 105,000 12 n number of terms a12 = 75,000 + (11)(15,000)

4 120,000 NA Sn sum of n terms

a12 = 75,000+165,000

15,000 d common difference

a12 = $240,000

17.
In the arithmetic sequence

4,7,10,13,…, which term has a

value of 301?

an a1 (n 1)d

301 4 ( n 1)3

301 4 3n 3

301 1 3n

300 3n

100 n

4,7,10,13,…, which term has a

value of 301?

an a1 (n 1)d

301 4 ( n 1)3

301 4 3n 3

301 1 3n

300 3n

100 n

18.
Try this one: Find n if an 633, a1 9, and d 24

9 a1 First term

633 an nth term

n n number of terms

NA Sn sum of n terms

24 d common difference

an a1 n 1 d

633 = 9 + (n - 1)(24)

633 = 9 + 24n - 24

633 = 24n – 15

648 = 24n

n = 27

9 a1 First term

633 an nth term

n n number of terms

NA Sn sum of n terms

24 d common difference

an a1 n 1 d

633 = 9 + (n - 1)(24)

633 = 9 + 24n - 24

633 = 24n – 15

648 = 24n

n = 27

19.
Given an arithmetic sequence with a15 38 and d 3, find a1.

a1 a1 First term

38 an nth term

15 n number of terms

NA Sn sum of n terms

-3 d common difference

an a1 n 1 d

38 = a1 + (15 - 1)(-3)

38 = a1 + (14)(-3)

38 = a1 - 42

a1 = 80

a1 a1 First term

38 an nth term

15 n number of terms

NA Sn sum of n terms

-3 d common difference

an a1 n 1 d

38 = a1 + (15 - 1)(-3)

38 = a1 + (14)(-3)

38 = a1 - 42

a1 = 80

20.
Find d if a1 6 and a 29 20

-6 a1 First term

20 an nth term

29 n number of terms

NA Sn sum of n terms

d d common difference

an a1 n 1 d

20 = -6 + (29 - 1)(d)

20 = -6 + (28)(d)

26 = 28d

13

d

14

-6 a1 First term

20 an nth term

29 n number of terms

NA Sn sum of n terms

d d common difference

an a1 n 1 d

20 = -6 + (29 - 1)(d)

20 = -6 + (28)(d)

26 = 28d

13

d

14

21.
Write an equation for the nth term of the arithmetic

sequence 8, 17, 26, 35, …

8 a1 First term

9 d common difference

an a1 n 1 d

an = 8 + (n - 1)(9)

an = 8 + 9n - 9

an = 9n - 1

sequence 8, 17, 26, 35, …

8 a1 First term

9 d common difference

an a1 n 1 d

an = 8 + (n - 1)(9)

an = 8 + 9n - 9

an = 9n - 1

22.
Arithmetic Mean: The terms between any two

nonconsecutive terms of an arithmetic sequence.

Ex. 19, 30, 41, 52, 63, 74, 85, 96

41, 52, 63 are the Arithmetic Mean between 30 and 74

nonconsecutive terms of an arithmetic sequence.

Ex. 19, 30, 41, 52, 63, 74, 85, 96

41, 52, 63 are the Arithmetic Mean between 30 and 74

23.
Find two arithmetic means between –4 and 5

-4, ____, ____, 5

-4 a1 First term an a1 n 1 d

5 an nth term 5 = -4 + (4 - 1)(d)

4 n number of terms 5 = -4 + (3)(d)

NA Sn sum of n terms 9 = (3)(d)

d d common difference d=3

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

forms an arithmetic sequence

-4, ____, ____, 5

-4 a1 First term an a1 n 1 d

5 an nth term 5 = -4 + (4 - 1)(d)

4 n number of terms 5 = -4 + (3)(d)

NA Sn sum of n terms 9 = (3)(d)

d d common difference d=3

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

forms an arithmetic sequence

24.
Find three arithmetic means between 21 and 45

21, ____, ____, ____, 45

21 a1 First term an a1 n 1 d

45 an nth term 45 = 21 + (5 - 1)(d)

5 n number of terms 45 = 21 + (4)(d)

NA Sn sum of n terms 24 = (4)(d)

d d common difference d=6

The three arithmetic means are 27, 33, and 39

since 21, 27, 33, 39, 45 forms an arithmetic sequence

21, ____, ____, ____, 45

21 a1 First term an a1 n 1 d

45 an nth term 45 = 21 + (5 - 1)(d)

5 n number of terms 45 = 21 + (4)(d)

NA Sn sum of n terms 24 = (4)(d)

d d common difference d=6

The three arithmetic means are 27, 33, and 39

since 21, 27, 33, 39, 45 forms an arithmetic sequence

25.
Arithmetic Series: An indicated sum of terms in an arithmetic

Arithmetic Sequence VS Arithmetic Series

3, 5, 7, 9, 11, 13 3 + 5 + 7 + 9 + 11 + 13

Arithmetic Sequence VS Arithmetic Series

3, 5, 7, 9, 11, 13 3 + 5 + 7 + 9 + 11 + 13

26.
Recall

Vocabulary of Sequences (Universal)

a1 First term

an nth term

n number of terms

Sn sum of n terms

d common difference

Vocabulary of Sequences (Universal)

a1 First term

an nth term

n number of terms

Sn sum of n terms

d common difference

27.
Find the sum of the first 63 terms of the arithmetic sequence -19, -13, -7,…

-19 a1 First term

353 ?? an nth term

63 n number of terms

Sn Sn sum of n terms

6 d common difference

n n

Sn a1 an an a1 n 1 d Sn a1 an

2 2

63

S63 19 353

2

a63 = 353

S63 10521

-19 a1 First term

353 ?? an nth term

63 n number of terms

Sn Sn sum of n terms

6 d common difference

n n

Sn a1 an an a1 n 1 d Sn a1 an

2 2

63

S63 19 353

2

a63 = 353

S63 10521

28.
Find the first 3 terms for an arithmetic series in which a1 = 9, an = 105, Sn =741.

9 a1 First term

105 an nth term

13 ?? n number of terms

9, 17, 25

741 Sn sum of n terms

?? d common difference

n

an a1 n 1 d Sn a1 an an a1 n 1 d

2

9 a1 First term

105 an nth term

13 ?? n number of terms

9, 17, 25

741 Sn sum of n terms

?? d common difference

n

an a1 n 1 d Sn a1 an an a1 n 1 d

2

29.
A radio station considered giving away $4000 every day in the month of August for a

total of $124,000. Instead, they decided to increase the amount given away every day

while still giving away the same total amount. If they want to increase the amount by

$100 each day, how much should they give away the first day?

a1 First term n

a1 Sn a1 an

2

?? an nth term

31 days n number of terms

$124,000 Sn sum of n terms

$100/day d common difference

n

Sn a1 an an a1 n 1 d

2

total of $124,000. Instead, they decided to increase the amount given away every day

while still giving away the same total amount. If they want to increase the amount by

$100 each day, how much should they give away the first day?

a1 First term n

a1 Sn a1 an

2

?? an nth term

31 days n number of terms

$124,000 Sn sum of n terms

$100/day d common difference

n

Sn a1 an an a1 n 1 d

2

30.
Sigma Notation ( )

Used to express a series or its sum in abbreviated form.

Used to express a series or its sum in abbreviated form.

31.
UPPER LIMIT

(NUMBER)

B

SIGMA

(SUM OF TERMS) a

nA

n NTH TERM

(SEQUENCE)

INDEX LOWER LIMIT

(NUMBER)

(NUMBER)

B

SIGMA

(SUM OF TERMS) a

nA

n NTH TERM

(SEQUENCE)

INDEX LOWER LIMIT

(NUMBER)

32.
4

j 2 1 2 2 2 3 2 4 2

j1

18

7

2a 2 4 2 5 2 6 2 7 44

a 4

If the sequence is arithmetic (has a common difference) you can use the Sn formula

4

n

j 2

1+2=3 a1 First term Sn a1 an

2

4+2=6 an nth term

4 n number of terms

?? Sn sum of n terms

NA d common difference

j 2 1 2 2 2 3 2 4 2

j1

18

7

2a 2 4 2 5 2 6 2 7 44

a 4

If the sequence is arithmetic (has a common difference) you can use the Sn formula

4

n

j 2

1+2=3 a1 First term Sn a1 an

2

4+2=6 an nth term

4 n number of terms

?? Sn sum of n terms

NA d common difference

33.
= 90

Is the sequence arithmetic?

10 + 17 + 26 + 37

No, there is no common difference

Thus you cannot use the Sn formula.

= 2.71828

Is the sequence arithmetic?

10 + 17 + 26 + 37

No, there is no common difference

Thus you cannot use the Sn formula.

= 2.71828

34.
Rewrite using sigma notation: 3 + 6 + 9 + 12

Arithmetic, d= 3

an a1 n 1 d

an 3 n 1 3

an 3n

4

3n

n1

Arithmetic, d= 3

an a1 n 1 d

an 3 n 1 3

an 3n

4

3n

n1