5-4 Sequences

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

5-4 Sequences

Warm UpFind the output for each input value.

Input Rule Output

–3x + 2 yx

–4

0

4

14

2

–10

5-4 Sequences

Problem of the Day

Function Rule A: Square the input. Divide by 2. Subtract 3. Function Rule B: Square the input. Subtract 6. Divide by 2.

If the input value for each rule is 222, what is the difference of the two output values? Why? 0; they are equivalent rules.

5-4 Sequences

Learn to find patterns to complete sequences using function tables.

5-4 Sequences

Vocabulary

sequence

term

arithmetic sequence

common difference

geometric sequence

5-4 Sequences

A sequence is an ordered list of numbers. Each number in a sequence is called a term.

When the sequence follows a pattern, the terms in the sequence are the output values of a function, and the value of each number depends on the number’s place in the list.

5-4 Sequences

You can use a variable such as n, to represent a number’s position in a sequence.

In an arithmetic sequence, the terms of the sequence differ by by the same nonzero number. This difference is called the common difference. In a geometric sequence, each term is multiplied by the same amount to get the next term in the sequence.

8642y (value of term)

4321n (position in the sequence)

5-4 Sequences

Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5.

Additional Example 1A: Identifying Patterns in a Sequence

-64-16-4-1y

54321n

In the sequence -1, -4, -16, -64, ,…, each number is multiplied by 4.

-256

-64 ● 4 = -256. Multiply the fourth number by 4.

The sequence is geometric. When n = 5, y = -256.

5-4 Sequences

Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5.

Additional Example 1B: Identifying Patterns in a Sequence

36414651y

54321n

31

In the sequence 51, 46, 41, 36, ,…, -5 is added each time.

36 + (-5) = 31. Add -5 to the fourth number.

The sequence is arithmetic. When n = 5, y = 31.

5-4 Sequences

Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5.

Check It Out: Example 1A

24201612y

54321n

28

In the sequence 12, 16, 20, 24, ,…, 4 is added each time.

24 + 4 = 28. Add 4 to the fourth number.

The sequence is arithmetic. When n = 5, y = 28.

5-4 Sequences

Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5.

Check It Out: Example 1B

-27-9-3-1y

54321n

In the sequence -1, -3, -9, -27, ,…, each number is multiplied by 3.

-81

-27 ● 3 = -81. Multiply the fourth number by 3.

The sequence is geometric. When n = 5, y = -81.

5-4 Sequences

Additional Example 2A: Identifying Functions in Sequences

Write a function that describes the sequence.

3, 6, 9, 12,…Make a function table.

124

93

62

31

yRulen

1 • 3

2 • 3

3 • 3

4 • 3

Multiply n by 3.

The function y = 3n describes this sequence.

5-4 Sequences

Additional Example 2B: Identifying Functions in Sequences

Write a function that describes the sequence.

4, 7, 10, 13,…Make a function table.

134

103

72

41

yRulen

3(1) + 1

3(2) + 1

3(3) + 1

3(4) + 1

Multiply n by 3 and add 1.

The function y = 3n + 1describes this sequence.

5-4 Sequences

Check It Out: Example 2A

Write a function that describes the sequence.

5, 6, 7, 8,…

Make a function table.

84

73

62

51

yRulen

1 + 4

2 + 4

3 + 4

4 + 4

Add 4 to n.

The function y = 4 + n describes this sequence.

5-4 Sequences

Check It Out: Example 2B

Write a function that describes the sequence.

3, 4, 5, 6,…

Make a function table.

610

59

48

37

yRulen

7 4

8 4

9 4

10 4

Subtract 4 from n.

The function y = n – 4 describes this sequence.

5-4 Sequences

Additional Example 3: Using Functions to Extend Sequences

Holli keeps a list showing her cumulative earnings for walking her neighbor’s dog. She recorded $1.25 the first time she walked the dog, $2.50 the second time, $3.75 the third time, and $5.00 the fourth time. Write a function that describes the sequence, and then use the function to predict her earnings after 9 walks.

Write the number of walks she recorded; 1.25, 2.50, 3.75, 5.00.

Make a function table.

5-4 Sequences

Additional Example 3 Continued

Multiply n by 1.25.

9 walks correspond to n = 9. When n = 9, y = 1.25 • 9 = 11.25. Holli would earn $11.25 after 9 walks.

y = 1.25n

5.004

3.753

2.502

1.251

yRulen

1 • 1.25

2 • 1.25

3 • 1.25

4 • 1.25

Write the function.

5-4 Sequences

Check It Out: Example 3

Jeff keeps a list showing his cumulative earnings for washing cars. He recorded $2.50 the first time he washed a car, $5 the second time, $7.50 the third time, and $10 the fourth time. Write a function that describes the sequence, and then use the function to predict his earnings after 8 car washes.

Write the number of car washed he recorded; 2.50, 5.00, 7.50, 10.00.

Make a function table.

5-4 Sequences

Check It Out: Example 3 Continued

Multiply n by 1.25.

8 car washed correspond to n = 8. When n = 8, y = 2.50 • 8 = 20. Jeff would earn $20. after 8 car washes.

y = 2.50n

10.004

7.503

5.002

2.501

yRulen

1 • 2.50

2 • 2.50

3 • 2.50

4 • 2.50

Write the function.

5-4 Sequences

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

5-4 Sequences

Lesson Quiz: Part I

Tell whether each sequence of y-values is arithmetic or geometric. Write a function that describes each sequence, and then find y when n = 5.

1. 6, 12, 18, 24,…

2. –3, –2, –1, 0,…

3. 24, 21, 18, 15,…

arithmetic; y = n – 4; 1

geometric; y = 6n; 30

arithmetic; y = 27 – 3n; 12

5-4 Sequences

Lesson Quiz: Part II

4. Arisha used 0.5 cups of nuts in the first batch of cookies that she made, 1 cup in the second, 1.5 cups in the third, and 2 cups in the fourth. Write a function to describe the sequence, and then use the function to predict the amount of nuts in the seventh batch of cookies.

y = 0.5n; 3.5 cups.

5-4 Sequences

1. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5.

4, 8, 12, 16, …

A. arithmetic; y = 4n; 20

B. geometric; y = 2n; 10

C. arithmetic; y = 4 + n; 9

D. arithmetic; y = 2n + 2; 12

Lesson Quiz for Student Response Systems

5-4 Sequences

2. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5.

–5, –4, –3, –2, …

A. arithmetic; y = n + 6; 11

B. geometric; y = 2n – 6; 4

C. arithmetic; y = n – 6; –1

D. geometric; y = n; 5

Lesson Quiz for Student Response Systems

5-4 Sequences

3. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5.

16, 12, 8, 4, …

A. arithmetic; y = 20 – 2n; 10

B. geometric; y = 4n; 20

C. arithmetic; y = 20 – 4n; 0

D. geometric; y = 30 – 6n; 0

Lesson Quiz for Student Response Systems

5-4 Sequences

4. Rita used 2 tablespoons of butter for the first recipe, 4 tablespoons for the second, 6 tablespoons for the third, and 8 tablespoons for the fourth. Write a function to describe the sequence, and then use the function to predict the amount of butter in the sixth recipe.

A. y = 2n; 12 tablespoons

B. y = 3n; 18 tablespoons

C. y = 4n; 24 tablespoons

D. y = 2n + 10; 22 tablespoons

Lesson Quiz for Student Response Systems