1. 4-5 Find a Pattern in Sequences

2. Essential Question
How do sequences help you to write a function?

3. Definitions term- each number in a sequence
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

4. n (position 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) 6 8

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

6. Example 2
Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. n 1 2 3 4 5 y 12 24 48 96
192
Multiply by 2 to the fourth number.
96 x 2 = 192
The sequence is geometric. When n = 5, y = 192.

7. 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

8. Example 3
Write a function that describes the sequence.
3, 6, 9, 12,… n
Rule y 1 3 2 6 9 4 12
1. Make a function table.
1 • 3
2 • 3
2. Multiply n by 3.
3 • 3
3. The function y = 3n describes this sequence.
4 • 3

9. Example 4
Write a function that describes the sequence.
4, 7, 10, 13,…
1. Make a function table. n
Rule y 1 4 2 7 3 10 13
3(1) + 1
3(2) + 1
2. Multiply n by 3 and add 1.
3(3) + 1
3(4) + 1
3. The function y = 3n + 1describes this sequence.

10. Example 5- Word Problem n Rule y 1 1.25 2 2.50 3 3.75 4 5.00
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. n
Rule y 1
1.25 2
2.50 3
3.75 4
5.00
2. Multiply n by 1.25.
1 • 1.25
3. y = 1.25n
2 • 1.25
3 • 1.25
4 • 1.25
9 walks correspond to n = 9. When n = 9, y = 1.25 • 9 = Holly would earn $11.25 after 9 walks.

11. Practice

12. Practice

13. Practice

14. Homework
Workbook pg. 40
All problems

15. 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,…
geometric; y = 6n; 30
arithmetic; y = n – 4; 1
arithmetic; y = 27 – 3n; 12