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4-5 Find a Pattern
in Sequences
Essential Question
How
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)
y (value of term)
1
2
3
4
2
4
6
8
Example 1
Tell whether the sequence of y-values is
arithmetic or geometric. Then find y when n = 5.
n
1
2
3
4
5
y
51
46
41
36
31
Subtract 5 to the fourth number.
36 - 5 = 31.
The sequence is arithmetic. When n = 5, y = 31.
Example 2
Tell whether the sequence of y-values is
arithmetic or geometric. Then find y when n = 5.
n
1
2
3
4
y
12
24
48
96
5
192
Multiply by 2 to the fourth number.
96 x 2 = 192
The sequence is geometric. When n = 5, y = 192.
Steps to finding the Function

1. Make a function table
 Column
1: n (position in sequence)
 Column 2: Rule
 Column 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 3
Write a function that describes the sequence.
3, 6, 9, 12,…
n
1
Rule
1•3
y
3
1. Make a function table.
2
2•3
6
2. Multiply n by 3.
3
3•3
9
4
4•3
12
3. The function y = 3n
describes this sequence.
Example 4
Write a function that describes the sequence.
4, 7, 10, 13,…
1. Make a function table.
n
1
Rule
3(1) + 1
y
4
2
3(2) + 1
7
3
3(3) + 1 10
4
3(4) + 1 13
2. Multiply n by 3 and
add 1.
3. The function y = 3n + 1describes this sequence.
Example 5- Word Problem
Holly 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.
3. y = 1.25n
n
1
Rule
y
1.25 1.25
1•
2
2 • 1.25 2.50
3
3 • 1.25 3.75
4
4 • 1.25 5.00
9 walks correspond to n = 9. When n = 9, y = 1.25 • 9
= 11.25. Holly would earn $11.25 after 9 walks.
Practice
Practice
Practice
Homework
Workbook
pg. 40
All 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,…
geometric; y = 6n; 30
2. –3, –2, –1, 0,…
arithmetic; y = n – 4; 1
3. 24, 21, 18, 15,… arithmetic; y = 27 – 3n; 12