9.2 – Arithmetic Sequences and Series - PowerPoint Presentation

Slide1

9.2 – Arithmetic Sequences and SeriesSlide2

An introduction…………

Arithmetic Sequences

ADD

To get next term

Geometric Sequences

MULTIPLY

To get next term

Arithmetic Series

Sum of Terms

Geometric Series

Sum of TermsSlide3

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

Arithmetic Sequence

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, 33Slide4

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, -32kSlide5

Vocabulary of Sequences (Universal)Slide6

Given an arithmetic sequence with

x

15

38

NA

-3

X = 80Slide7

-19

63

??

x

6

353Slide8

Try this one:

1.5

16

x

NA

0.5Slide9

9

x

633

NA

24

X = 27Slide10

-6

29

20

NA

xSlide11

Find two arithmetic means between –4 and 5

-4, ____, ____, 5

-4

4

5

NA

x

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

-1, 2

, 5

forms an arithmetic sequenceSlide12

Find three arithmetic means between 1 and 4

1, ____, ____, ____, 4

1

5

4

NA

x

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

since 1,

7/4, 10/4, 13/4

, 4 forms an arithmetic sequenceSlide13

Find n for the series in which

5

x

y

440

3

X = 16

Graph on positive windowSlide14

Example: The nth Partial Sum

The

sum

of the first n terms of an infinite sequence is called the

nth partial sum. Slide15

Example 6. Find the 150

th

partial sum of the arithmetic sequence, 5, 16, 27, 38, 49, …Slide16

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?Slide17

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.

So the total sales for the first 2o years is $1,625,000Slide18

9.3 – Geometric Sequences and SeriesSlide19

Arithmetic Sequences

ADD

To get next term

Geometric Sequences

MULTIPLY

To get next term

Arithmetic Series

Sum of Terms

Geometric Series

Sum of TermsSlide20

Vocabulary of Sequences (Universal)Slide21

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

3 – 2 vs. 9/2 – 3… not arithmeticSlide22

1/2

x

9

NA

2/3Slide23

Find two geometric means between –2 and 54

-2, ____, ____, 54

-2

54

4

NA

x

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

6, -18

, 54

forms an geometric sequenceSlide24

-3, ____, ____, ____Slide25

x

9

NASlide26

x

5

NASlide27

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

*** denotes trick question

1/4

3

NASlide28

1/2

7

xSlide29

Section 12.3 – Infinite SeriesSlide30

1, 4, 7, 10, 13, ….

Infinite Arithmetic

No Sum

3, 7, 11, …, 51

Finite Arithmetic

1, 2, 4, …, 64

Finite Geometric

1, 2, 4, 8, …

Infinite Geometric

r

>

1

r < -1

No Sum

Infinite Geometric

-1

<

r

<

1Slide31

Find the sum, if possible: Slide32

Find the sum, if possible: Slide33

Find the sum, if possible: Slide34

Find the sum, if possible: Slide35

Find the sum, if possible: Slide36

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

32/5Slide37

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

225/4Slide38

Sigma NotationSlide39

UPPER BOUND

(NUMBER)

LOWER BOUND

(NUMBER)

SIGMA

(SUM OF TERMS)

NTH TERM

(SEQUENCE)Slide40
Slide41
Slide42
Slide43

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

Arithmetic, d= 3Slide44

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

Geometric, r = ½ Slide45

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

Not Arithmetic, Not Geometric

19 + 18 + 16 + 12 + 4

-1 -2 -4 -8Slide46

Rewrite the following using sigma notation:

Numerator is geometric, r = 3

Denominator is arithmetic d= 5

NUMERATOR:

DENOMINATOR:

SIGMA NOTATION: