4-5 Find a Pattern in Sequences. Essential Question How do sequences help you to write a function?

4-5 Find a Pattern

in Sequences

Essential QuestionHow do sequences help

you to write a function?

Definitions• sequence- an ordered list of numbers• term- each number in a sequence • arithmetic sequence- the same value is added each time to get the next term in the sequence• geometric sequence- each term is multiplied by the same value to get the next term in the sequence

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.

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

n (position in the sequence)

1 2 3 4

y (value of term) 2 4 6 8

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

Example 1

n 1 2 3 4 5

y 51 46 41 36 31

36 - 5 = 31.

Subtract 5 to the fourth number.

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

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

Example 2

n 1 2 3 4 5

y 12 24 48 96 192

96 x 2 = 192

Multiply by 2 to the fourth number.

The sequence is geometric. When n = 5, y = 192.

Steps to finding the Function

1. Make a function tableColumn 1: n (position in sequence)Column 2: RuleColumn 3: y (values given in problem)

2. Find the Rule by deciding what it takes to get from n to y

3. Write the function

Example 3Write a function that describes the sequence.

3, 6, 9, 12,…

1. Make a function table.n Rule y

1 3

2 6

3 9

4 12

1 • 3

2 • 3

3 • 3

4 • 3

2. Multiply n by 3.

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

Example 4Write a function that describes the sequence.

4, 7, 10, 13,…

1. Make a function table.

n Rule y

1 4

2 7

3 10

4 13

3(1) + 1

3(2) + 1

3(3) + 1

3(4) + 1

2. Multiply n by 3 and add 1.

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

Example 5- Word ProblemHolly 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.

1. Make a function table.

2. Multiply n by 1.25.

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

3. y = 1.25n

n Rule y

1 1.25

2 2.50

3 3.75

4 5.00

1 • 1.25

2 • 1.25

3 • 1.25

4 • 1.25

Practice

Practice

Practice

Homework

Workbook pg. 40All problems

Closing

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