Arithmetic Sequences and Series

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

Geometric Sequences and Series

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

Vocabulary of Sequences (Universal)

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

n 1

n 1

n1

n

nth term of geometric sequence

sum of n terms of geometric sequ

a a r

a r 1S

r 1ence

Find the next three terms of 2, 3, 9/2, ___, ___, ___

3 – 2 vs. 9/2 – 3… not arithmetic3 9 / 2 3

1.5 geometric r2 3 2

3 3 3 3 3 3

2 2 2

92, 3, , , ,

2

9 9 9

2 2 2 2 2 2

92, 3, , ,

27 81 243

4 8,

2 16

1 9

1 2If a , r , find a .

2 3

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/2

x

9

NA

2/3

n 1n 1a a r

9 11 2

x2 3

8

8

2x

2 3

7

8

2

3 128

6561

Find two geometric means between –2 and 54

-2, ____, ____, 54

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

-2

54

4

NA

x

n 1n 1a a r

1454 2 x

327 x 3 x

The two geometric means are 6 and -18, since –2, 6, -18, 54

forms an geometric sequence

2 4 1

2Find a a if a 3 and r

3

-3, ____, ____, ____

2Since r ...

3

4 83, 2, ,

3 9

2 4

8 10a a 2

9 9

9Find a of 2, 2, 2 2,...

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

x

9

NA

2

2 2 2r 2

22

n 1n 1a a r

9 1

x 2 2

8

x 2 2

x 16 2

5 2If a 32 2 and r 2, find a

____, , ____,________ ,32 2

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

x

5

NA

32 2

2n 1

n 1a a r

5 1

32 2 x 2

4

32 2 x 2

32 2 x4

8 2 x

*** Insert one geometric mean between ¼ and 4***

*** denotes trick question

1,____,4

4

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/4

3

NA

4

xn 1

n 1a a r

3 114

4r 2r

14

4 216 r 4 r

1,1, 4

4

1, 1, 4

4

7

1 1 1Find S of ...

2 4 8

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/2

7

x

NA

11184r

1 1 22 4

n1

n

a r 1S

r 1

71 12 2

x12

1

1

71 12 2

12

1

63

64

Section 12.3 – Infinite Series

1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum

3, 7, 11, …, 51 Finite Arithmetic n 1 n

nS a a

2

1, 2, 4, …, 64 Finite Geometric n

1

n

a r 1S

r 1

1, 2, 4, 8, … Infinite Geometricr > 1r < -1

No Sum

1 1 13,1, , , ...

3 9 27Infinite Geometric

-1 < r < 11a

S1 r

Find the sum, if possible: 1 1 1

1 ...2 4 8

1 112 4r

11 22

1 r 1 Yes

1a 1S 2

11 r 12

Find the sum, if possible: 2 2 8 16 2 ...

8 16 2r 2 2

82 2 1 r 1 No

NO SUM

Find the sum, if possible: 2 1 1 1

...3 3 6 12

1 113 6r

2 1 23 3

1 r 1 Yes

1

2a 43S

11 r 312

Find the sum, if possible: 2 4 8

...7 7 7

4 87 7r 22 47 7

1 r 1 No

NO SUM

Find the sum, if possible: 5

10 5 ...2

55 12r

10 5 2 1 r 1 Yes

1a 10S 20

11 r 12

The Bouncing Ball Problem – Version A

A ball is dropped from a height of 50 feet. It rebounds 4/5 of

it’s height, and continues this pattern until it stops. How far

does the ball travel?50

40

32

32/5

40

32

32/5

40S 45

504

10

1554

The Bouncing Ball Problem – Version B

A ball is thrown 100 feet into the air. It rebounds 3/4 of

it’s height, and continues this pattern until it stops. How far

does the ball travel?

100

75

225/4

100

75

225/4

10S 80

100

4 43

1

0

10

3

Sigma Notation

B

nn A

a

UPPER BOUND(NUMBER)

LOWER BOUND(NUMBER)

SIGMA(SUM OF TERMS) NTH TERM

(SEQUENCE)

j

4

1

j 2

21 2 2 3 2 24 18

7

4a

2a 42 2 5 2 6 72 44

n

n 0

4

0.5 2

00.5 2 10.5 2 20.5 2 30.5 2 40.5 2

33.5

0

n

b

36

5

0

36

5

13

65

23

65

...

1aS

1 r

6

153

15

2

x

3

7

2x 1

2 1 2 8 1 2 9 1 ...7 2 123

n 1 n

2n 1S a a 15

2

3

2

747

527

1

b

9

4

4b 3

4 3 4 5 3 4 6 3 ...4 4 319

n 1 n

1n 1S a a 19

2

9

2

479

784

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

Arithmetic, d= 3

n 1a a n 1 d

na 3 n 1 3

na 3n4

1n

3n

Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1

Geometric, r = ½ n 1

n 1a a r n 1

n

1a 16

2

n 1

n

5

1

116

2

Rewrite using sigma notation: 19 + 18 + 16 + 12 + 4

Not Arithmetic, Not Geometric

n 1na 20 2

n 1

n

5

1

20 2

19 + 18 + 16 + 12 + 4 -1 -2 -4 -8

Rewrite the following using sigma notation:3 9 27

...5 10 15

Numerator is geometric, r = 3Denominator is arithmetic d= 5

NUMERATOR: n 1

n3 9 27 ... a 3 3

DENOMINATOR: n n5 10 15 ... a 5 n 1 5 a 5n

SIGMA NOTATION: 1

1

n

n 5n

3 3