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

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

Vocabulary sequence term arithmetic sequence geometric sequence

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.

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.

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.

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.

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. 24 + 4 = 28. Add 4 to the fourth number. The sequence is arithmetic. When n = 5, y = 28.

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.

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.

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.

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.

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.

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.

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 = 11.25. Holli would earn $11.25 after 9 walks.

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.

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.

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,…

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. .