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12-2 Find a Pattern in Sequences Warm Up Problem of the Day Lesson Presentation Course 2 12-2 Find a Pattern in Sequences Warm Up Find the output for each input value. Course 2 Input Rule Output x –3x + 2 y –4 14 0 2 4 –10 12-2 Find a Pattern in 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. Course 2 12-2 Find a Pattern in Sequences Learn to find patterns to complete sequences using function tables. Course 2 12-2 Insert Lesson Title Here Find a Pattern in Sequences Vocabulary sequence Course 2 12-2 Find a Pattern in Sequences A sequence is an ordered list of numbers. One of the most well-known sequences is the Fibonacci sequence. In this sequence, each term after the second term is the sum of the two terms before it. 1, 1, 2, 3, 5, 8, 13,… When the list follows a pattern, the numbers 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. Course 2 12-2 Find a Pattern in Sequences You can use a variable to represent a number’s place in a sequence. n (place in list) 1st number 2nd number 3rd number 4th number Number in sequence: You can use a function table to help identify the pattern in a sequence. Course 2 12-2 Find a Pattern in Sequences Additional Example 1A: Identifying Functions in a Sequence Find a function that describes each sequence. Use y for the term in the sequence and n for its place in the list. Then use the function to find the next three terms in the sequence. A. 4, 8, 12, 16, . . . n Rule y 1 1·4 4 2 2·4 8 3 3·4 4·4 12 4 Course 2 16 Multiply n by 4. 12-2 Find a Pattern in Sequences Additional Example 1A Continued Use the function to find the next three terms in the sequence. A. 4, 8, 12, 16, . . . y = 4n Write the function. y = 4(5) = 20 Substitute for n to find the next three terms. y = 4(6) = 24 y = 4(7) = 28 Course 2 12-2 Find a Pattern in Sequences Additional Example 1B: Identifying Functions in Sequences Find a function that describes the sequence. Use y for the term in the sequence and n for its place in the list. Then use the function to find the next three terms in the sequence. B. –4, –3, –2, –1, . . . n Rule y 1 1–5 –4 2 2–5 –3 3 3–5 4–5 –2 4 Course 2 –1 Subtract 5 from n. 12-2 Find a Pattern in Sequences Additional Example 1B Continued Use the function to find the next three numbers in the sequence. B. –4, –3, –2, –1, . . . y=n–5 Write the function. y = (5) – 5 = 0 Substitute for n to find the next three terms. y = (6) – 5 = 1 y = (7) – 5 = 2 Course 2 12-2 Insert Here Find a Lesson Pattern Title in Sequences Try This: Example 1 Find a function that describes each sequence. Use y for the term in the sequence and n for its place in the list. Then use the function to find the next three terms in the sequence. A. 3, 6, 9, 12, . . . n Rule y 1 1·3 3 2 2·3 6 3 3·3 4·3 9 4 Course 2 12 Multiply n by 3. 12-2 Insert Here Find a Lesson Pattern Title in Sequences Try This: Example 1A Continued Use the function to find the next three terms in the sequence. A. 3, 6, 9, 12, . . . y = 3n Write the function. y = 3 (5) = 15 Substitute for n to find the next three numbers y = 3 (6) = 18 y = 3 (7) = 21 Course 2 12-2 Insert Here Find a Lesson Pattern Title in Sequences Try This: Example 1B Find a function that describes the sequence. Use y for the term in the sequence and n for its place in the list. Then use the function to find the next three terms in the sequence. B. –8, –7, –6, –5, . . . n Rule y 1 1–9 –8 2 2–9 –7 3 3–9 4–9 –6 4 Course 2 –5 Subtract 9 from n. 12-2 Find a Pattern in Sequences Try This: Example 1B Continued Use the function to find the next three numbers in the sequence. B. –8, –7, –6, –5, . . . y=n–9 Write the function. y = (5) – 9 = –4 Substitute for n to find the next three terms. y = (6) – 9 = –3 y = (7) – 9 = –2 Course 2 12-2 Find a Pattern in Sequences Additional Example 2: Problem Solving Application Holli earned $3.50 a day pet sitting for her neighbor’s fish. Holli pet sat for 5 days. Her neighbor also paid her $8 for picking up the mail. Write a sequence showing how much money Holli earned after 1, 2, 3, 4, and 5 days. Course 2 Find a Pattern in Sequences 12-2 Additional Example 2 Continued 1 Understand the Problem List the important information: • Holly earns $3.50 each day. • She pet sits for 5 days. • She gets paid $8 for picking up the mail. The answer will be the amount of money she has earned after 1, 2, 3, 4, and 5 days. Course 2 Find a Pattern in Sequences 12-2 Additional Example 2 Continued 2 Make a Plan You can find a pattern, then write a rule that can be used to find a sequence. After the first day, Holli will have earned $3.50 + $8. After the second day, Holli will have earned 2 · $3.50 + $8. The rule n · $3.50 + $8 can be used to find the terms in the sequence. Course 2 12-2 Find a Pattern in Sequences Additional Example 2 Continued 3 Solve n Rule y 1 1 · 3.50 + 8 11.50 2 2 · 3.50 + 8 15.00 3 3 · 3.50 + 8 18.50 4 4 · 3.50 + 8 22.00 5 5 · 3.50 + 8 25.50 After 1, 2, 3, 4, and 5 days, Holli will have earned $11.50, $15.00, $18.50, $22.00, and $25.50, respectively. Course 2 12-2 Find a Pattern in Sequences Additional Example 2 Continued 4 Look Back If Holli earns $4 per day, she will have earned about $12, $16, $20, $24, and $28 after each of the 5 days respectively. The answer is reasonable. Course 2 12-2 Insert Here Find a Lesson Pattern Title in Sequences Try This: Example 2 Bob spends $12.50 each week for a lunch ticket. In addition, he usually spends $6.50 a day for incidentals. Write a sequence showing how much money Bob had spent after 1, 2, 3, 4, and 5 days. Course 2 12-2 Insert Here Find a Lesson Pattern Title in Sequences Try This: Example 2 Continued 1 Understand the Problem List the important information: • Bob spends $12.50 each week for meals. • He spends $6.50 a day on incidentals. The answer will be the amount of money he has spent after 5 days. Course 2 12-2 Insert Here Find a Lesson Pattern Title in Sequences Try This: Example 2 Continued 2 Make a Plan You can find a pattern, then write a rule that can be used to find a sequence. After the first day, Bob will have spent $6.50 + $12.50. After the second day, Bob will have spent 2 · $6.50 + $12.50. The rule n · $6.50 + $12.50 can be used to find the terms in the sequence. Course 2 Insert Here Find a Lesson Pattern Title in Sequences 12-2 3 Try This: Example 2 Continued Solve n Rule y 1 1 · 6.50 + 12.5 19.00 2 2 · 6.50 + 12.5 25.50 3 3 · 6.50 + 12.5 32.00 4 4 · 6.50 + 12.5 38.50 5 5 · 6.50 + 12.5 45.00 After 1, 2, 3, 4 and 5 days, Bob will have spent $19.00, $25.50, $32.00, $38.50, and $45.00, respectively. Course 2 12-2 Insert Here Find a Lesson Pattern Title in Sequences Try This: Example 2 Continued 4 Look Back If Bob spends 6.50 per day, he will have spent $19.00, $25.50, $32.00, $38.50, and $45.00 after each of the 5 days respectively. The answer is reasonable. Course 2 12-2 Find Insert a Pattern Lesson Title in Sequences Here Lesson Quiz: Part 1 Find a function that describes each sequence. Use the function to find the next three terms in the sequence. 1. 6, 12, 18, 24 y = 6n; 30, 36, 42 2. –3, –2, –1, 0 y = n – 4; 1, 2, 3 3. 24, 21, 18, 15 y = 27 – 3n; 12, 9, 6 4. 7, 14, 21, 28 y = 7n; 35, 42, 49 Course 2 12-2 Find Insert a Pattern Lesson Title in Sequences Here Lesson Quiz: Part 2 5. Arisha bought a 5-pound bag of chocolate chips. She adds 0.5 lb of chips to each batch of cookies she makes. Find a function that describes this relationship. How many pounds of chips remain after she makes 5 batches of cookies? y = 5 – 0.5x; 2.5 lb remain. Course 2