1. 5-4 Sequences Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes

2. 5-4 Sequences Warm Up Find the output for each input value. Input RuleOutput –3x + 2yx –4 0 4 14 2 –10

3. 5-4 Sequences Problem of the Day Function Rule A: Square the input. Divide by 2. Subtract 3. Function Rule B: Square the input. Subtract 6. Divide by 2. If the input value for each rule is 222, what is the difference of the two output values? Why? 0; they are equivalent rules.

4. 5-4 Sequences Learn to find patterns to complete sequences using function tables.

5. 5-4 Sequences Vocabulary sequence term arithmetic sequence common difference geometric sequence

6. 5-4 Sequences 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.

7. 5-4 Sequences You can use a variable such as n, to represent a number’s position in a sequence. In an arithmetic sequence, the terms of the sequence differ by by the same nonzero number. This difference is called the common difference. In a geometric sequence, each term is multiplied by the same amount to get the next term in the sequence. 8642y (value of term) ‏ 4321n (position in the sequence) ‏

8. 5-4 Sequences Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. Additional Example 1A: Identifying Patterns in a Sequence -64-16-4y 54321n In the sequence -1, -4, -16, -64,,…, each number is multiplied by 4. -256 -64 ● 4 = -256.Multiply the fourth number by 4. The sequence is geometric. When n = 5, y = -256.

9. 5-4 Sequences Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. Additional Example 1B: Identifying Patterns in a Sequence 36414651y 54321n 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.

10. 5-4 Sequences Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. Check It Out: Example 1A 24201612y 54321n 28 In the sequence 12, 16, 20, 24,,…, 4 is added each time. 24 + 4 = 28.Add 4 to the fourth number. The sequence is arithmetic. When n = 5, y = 28.

11. 5-4 Sequences Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. Check It Out: Example 1B -27-9-3y 54321n In the sequence -1, -3, -9, -27,,…, each number is multiplied by 3. -81 -27 ● 3 = -81.Multiply the fourth number by 3. The sequence is geometric. When n = 5, y = -81.

12. 5-4 Sequences Additional Example 2A: Identifying Functions in Sequences Write a function that describes the sequence. 3, 6, 9, 12,… Make a function table. 124 93 62 31 yRulen 1 3 2 3 3 4 3 Multiply n by 3. The function y = 3n describes this sequence.

13. 5-4 Sequences Additional Example 2B: Identifying Functions in Sequences Write a function that describes the sequence. 4, 7, 10, 13,… Make a function table. 134 103 72 41 yRulen 3(1) + 1 3(2) + 1 3(3) + 1 3(4) + 1 Multiply n by 3 and add 1. The function y = 3n + 1describes this sequence.

14. 5-4 Sequences Check It Out: Example 2A Write a function that describes the sequence. 5, 6, 7, 8,… Make a function table. 84 73 62 51 yRulen 1 + 4 2 + 4 3 + 4 4 + 4 Add 4 to n. The function y = 4 + n describes this sequence.

15. 5-4 Sequences Check It Out: Example 2B Write a function that describes the sequence. 3, 4, 5, 6,… Make a function table. 610 59 48 37 yRulen 7  4 8  4 9  4 10  4 Subtract 4 from n. The function y = n – 4 describes this sequence.

16. 5-4 Sequences 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.

17. 5-4 Sequences Additional Example 3 Continued Multiply n by 1.25. 9 walks correspond to n = 9. When n = 9, y = 1.25 9 = 11.25. Holli would earn $11.25 after 9 walks. y = 1.25n 5.004 3.753 2.502 1.251 yRulen 1 1.25 2 1.25 3 1.25 4 1.25 Write the function.

18. 5-4 Sequences 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, 10.00. Make a function table.

19. 5-4 Sequences Check It Out: Example 3 Continued Multiply n by 1.25. 8 car washed correspond to n = 8. When n = 8, y = 2.50 8 = 20. Jeff would earn $20. after 8 car washes. y = 2.50n 10.004 7.503 5.002 2.501 yRulen 1 2.50 2 2.50 3 2.50 4 2.50 Write the function.

20. 5-4 Sequences Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

21. 5-4 Sequences Lesson Quiz: Part I 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

22. 5-4 Sequences Lesson Quiz: Part II 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. y = 0.5n; 3.5 cups.

23. 5-4 Sequences 1. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5. 4, 8, 12, 16, … A. arithmetic; y = 4n; 20 B. geometric; y = 2n; 10 C. arithmetic; y = 4 + n; 9 D. arithmetic; y = 2n + 2; 12 Lesson Quiz for Student Response Systems

24. 5-4 Sequences 2. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5. –5, –4, –3, –2, … A. arithmetic; y = n + 6; 11 B. geometric; y = 2n – 6; 4 C. arithmetic; y = n – 6; –1 D. geometric; y = n; 5 Lesson Quiz for Student Response Systems

25. 5-4 Sequences 3. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5. 16, 12, 8, 4, … A. arithmetic; y = 20 – 2n; 10 B. geometric; y = 4n; 20 C. arithmetic; y = 20 – 4n; 0 D. geometric; y = 30 – 6n; 0 Lesson Quiz for Student Response Systems

26. 5-4 Sequences 4. Rita used 2 tablespoons of butter for the first recipe, 4 tablespoons for the second, 6 tablespoons for the third, and 8 tablespoons for the fourth. Write a function to describe the sequence, and then use the function to predict the amount of butter in the sixth recipe. A. y = 2n; 12 tablespoons B. y = 3n; 18 tablespoons C. y = 4n; 24 tablespoons D. y = 2n + 10; 22 tablespoons Lesson Quiz for Student Response Systems