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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