1. Week one of the first six weeks
4-5
Find a Pattern in Sequences
Course 2
Week one of the first six weeks
7th grade and 7th a/p

2. Learning objective: to find patterns to complete sequences using function tables.

3. Vocabulary
sequence
term
arithmetic sequence
geometric sequence

4. A sequence is an ordered list of numbers
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. n (position in the sequence)
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
In an arithmetic sequence, the same value is added each time to get the next term in the sequence. In a geometric sequence, each term is multiplied by the same value to get the next term in the sequence.

6. Additional Example 1A: Identifying Patterns in a Sequence
Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. n 1 2 3 4 5 y -1 -4
-16
-64
-256
In the sequence -1, -4, -16, -64, ,…, each number is multiplied by 4.
-64 ● 4 = -256.
Multiply the fourth number by 4.
The sequence is geometric. When n = 5, y = -256.

7. Additional Example 1B: Identifying Patterns in a Sequence
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
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.

8. Check It Out: Example 1A
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 16 20 24 28
In the sequence 12, 16, 20, 24, ,…, 4 is added each time.
= 28.
Add 4 to the fourth number.
The sequence is arithmetic. When n = 5, y = 28.

9. n 1 2 3 4 5 y -1 -3 -9 -27 Check It Out: Example 1B
Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. n 1 2 3 4 5 y -1 -3 -9
-27
-81
In the sequence -1, -3, -9, -27, ,…, each number is multiplied by 3.
-27 ● 3 = -81.
Multiply the fourth number by 3.
The sequence is geometric. When n = 5, y = -81.

10. Additional Example 2A: Identifying Functions in Sequences
Write a function that describes the sequence.
3, 6, 9, 12,…
Make a function table. n
Rule y 1 3 2 6 9 4 12
1 • 3
2 • 3
Multiply n by 3.
3 • 3
4 • 3
The function y = 3n describes this sequence.

11. Additional Example 2B: Identifying Functions in Sequences
Write a function that describes the sequence.
4, 7, 10, 13,…
Make a function table. n
Rule y 1 4 2 7 3 10 13
3(1) + 1
3(2) + 1
Multiply n by 3 and add 1.
3(3) + 1
3(4) + 1
The function y = 3n + 1describes this sequence.

12. Check It Out: Example 2A
Write a function that describes the sequence.
5, 6, 7, 8,…
Make a function table. n
Rule y 1 5 2 6 3 7 4 8
1 + 4
2 + 4
Add 4 to n.
3 + 4
4 + 4
The function y = 4 + n describes this sequence.

13. Check It Out: Example 2B
Write a function that describes the sequence.
3, 4, 5, 6,…
Make a function table. n
Rule y 7 3 8 4 9 5 10 6
7 - 4
8 - 4
Subtract 4 from n.
9 - 4
10 - 4
The function y = n – 4 describes this sequence.

14. 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.

15. Additional Example 3 Continued
Rule y 1
1.25 2
2.50 3
3.75 4
5.00
1 • 1.25
Multiply n by 1.25.
2 • 1.25
3 • 1.25
4 • 1.25
Write the function.
y = 1.25n
9 walks correspond to n = 9. When n = 9, y = 1.25 • 9 = Holli would earn $11.25 after 9 walks.

16. 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,
Make a function table.

17. Check It Out: Example 3 Continued
Rule y 1
2.50 2
5.00 3
7.50 4
10.00
1 • 2.50
Multiply n by 1.25.
2 • 2.50
3 • 2.50
4 • 2.50
Write the function.
y = 2.50n
8 car washed correspond to n = 8. When n = 8, y = 2.50 • 8 = 20. Jeff would earn $20. after 8 car washes.

18. Demonstration of Learning
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,…

19. Demonstration of learning continued
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. .