Transcription of Name Class Date 5-1 - MS. FATIMA'S WEBSITE
1 Houghton Mifflin Harcourt Publishing CompanyName Class date 5-1 Video TutorUnderstanding SequencesA sequence is an ordered list of numbers or other items. Each element in a sequence is called a term. For instance, in the sequence 1, 3, 5, 7, 9, .., the second term is term in a sequence can be paired with a position number, and these pairings establish a function whose domain is the set of position numbers and whose range is the set of terms, as illustrated below. The position numbers are consecutive integers that typically start at either 1 or the sequence shown in the table, you can write f (4) = 7, which can be interpreted as the fourth term of the sequence is 7. REFLECT1a. The domain of the function f defining the sequence 2, 5, 8, 11, 14, .. is the set of consecutive integers starting with 0. What is f (4)?
2 Explain how you determined your answer. 1b. How does your answer to Question 1a change if the domain of the function is the set of consecutive integers starting with 1? 1c. Predict the next term in the sequence 48, 42, 36, 30, 24, .. Explain your reasoning. 1d. Why is the relationship between the position numbers and the terms of a sequence a function? 1e. Give an example of a sequence from your everyday life. Explain why your example represents a sequence. ENGAGE1 Introduction to SequencesGoing DeeperEssential question: Why is a sequence a function?Position numbern12345 DomainTerm of sequencef(n)13579 RangeMCC9 5 135 Lesson 1 Houghton Mifflin Harcourt Publishing CompanySome numerical sequences can be described by using algebraic rules. An explicit rule for a sequence defines the nth term as a function of an Explicit Rule to Generate a SequenceWrite the first 4 terms of the sequence f(n) = n 2 + 1.
3 Assume that the domain of the function is the set of consecutive integers starting with first 4 terms are .REFLECT2a. How could you use a graphing calculator to check your answer?2b. Explain how to find the 20th term of the recursive rule for a sequence defines the nth term by relating it to one or more previous a Recursive Rule to Generate a SequenceWrite the first 4 terms of the sequence with f(1) = 3 and f(n) = f(n - 1) + 2 for n 2. Assume that the domain of the function is the set of consecutive integers starting with first term is given: f (1) = 3. Use f (1) to find f (2), f (2) to find f (3), and so on. In general, f (n - 1) refers to the term that precedes f (n).The first 4 terms are .EXAMPLE2 EXAMPLE3n n 2 + 1f(n)1 2 + 1 = + 12 2 + 1 = + 13 2 + 1 = + 14 2 + 1 = + 1nf(n - 1) + 2f(n)2f(2 - 1) + 2 = f(1) + 2 = 3 + 23f ( - 1 ) + 2 = f ( ) + 2 = + 24f ( - 1 ) + 2 = f ( ) + 2 = + 2 MCC9 5 136 Lesson 1 Houghton Mifflin Harcourt Publishing CompanyREFLECT3a.
4 Describe how to find the 12th term of the Suppose you want to find the 50th term of a sequence. Would you rather use a recursive rule or an explicit rule? Explain your a SequenceA male honeybee has one female parent, and a female honeybee has one male and one female parent. In the diagram below, a male honeybee is represented by M in row 1. His parent is represented by F in row 2. Her parents are represented by M and F in row 3, and so on. Write a recursive rule for a sequence that describes the number of bees in each Extend the diagram to show rows 5, 6, and Complete the table to show the number of bees in each (position number)1234567 Number of bees (term of sequence)112 MCC9 1 Row 2 Row 6 Row 5 Row 4 Row 3 Row 7 Module 5 137 Lesson 1 Houghton Mifflin Harcourt Publishing CompanyC Write a recursive rule for the sequence in the table.
5 Assume that the domain of the function is the set of consecutive integers starting with , write the rule in words. The first two terms are both . Every other term is the of the previous two , write the rule algebraically. f (1) = f (2) = and The first and second terms are both 1. f (n) = f ( n - ) + f (n - 2) for n Each successive term is the sum of the preceding two If you continued the pattern in the diagram, how many bees would be in the 8th row? Explain how you determined your answer. 4b. The sequence given in the table, 1, 1, 2, 3, 5, 8, 13, .., is called the Fibonacci sequence. An explicit rule for the Fibonacci sequence is f (n) = 1 ____ 5 ( 1 + 5 _____ 2 ) n - 1 ____ 5 ( 1 - 5 _____ 2 ) n where the values of n are consecutive integers starting with 1.
6 Use the explicit rule to show that f (1) = 1. Then use a calculator and the explicit rule to find the 9th term of the Fibonacci Now use the recursive rule to find the 9th term of the Fibonacci sequence. Does your result agree with the result from the explicit rule? 4d. Which rule for the Fibonacci sequence would be easier to use if you did not have a calculator? The number of petals on many flowers is equal to a Fibonacci number, that is, one of the terms in the Fibonacci sequence. Based on this fact, is a flower more likely to have 20 petals or 21 petals? 5 138 Lesson 1 Houghton Mifflin Harcourt Publishing CompanyPrActicEWrite the first four terms of each sequence. Assume that the domain of the function is the set of consecutive integers starting with 1. 5. f (1) = 2 and f (n) = f (n - 1) + 10 for n 2 6.
7 F (1) = 16 and f (n) = 1 __ 2 f (n - 1) for n 2 7. f (1) = 1 and f (n) = 2 f (n - 1) + 1 for n 2 8. f (1) = f (2) = 1 and f (n) = f (n - 2) - f (n - 1) for n 3 9. Each year for the past 4 years, Donna has gotten a raise equal to 5% of the previous year s salary. Her starting salary was $40,000. a. Complete the table to show Donna s salary over time. b. Write a recursive rule for the sequence in the table. Assume that the domain of the function is the set of consecutive integers starting with 0, so the first term of the sequence is f (0). c. What is f (7), rounded to the nearest whole number? What does f (7) represent in this situation?Write the 12th term of each sequence. Assume that the domain of the function is the set of consecutive integers starting with 1. 1. f (n) = (n - 1) 2 3.
8 F (n) = 4(0. 5) n 2. f (n) = n + 1 _____ n + 3 4. f (n) = n - 1 10. f (n) = 3n - 2 11. f (n) = 2n(n + 1) Year (position number)Salary ($) (term of sequence)040,0001234 Module 5 139 Lesson 1 Houghton Mifflin Harcourt Publishing Company 12. The diagram shows the first four figures in a pattern of dots. a. Draw the next figure in the pattern. b. Use the pattern to complete the table. c. Write an explicit rule for the sequence in the table. Assume that the domain of the function is the set of consecutive integers starting with 1. d. How many dots will be in the 10th figure of the pattern? Figure (position number)Number of dots (term of sequence)112345nf(n)16273849510nf(n)1829 310411512nf(n)132639412515nf(n)122438416 532nf(n)112 1 __ 2 3 1 __ 3 4 1 __ 4 5 1 __ 5 nf(n)127224321418515 Write an explicit rule for each sequence.
9 Assume that the domain of the function is the set of consecutive integers starting with a recursive rule for each sequence. Assume that the domain of the function is the set of consecutive integers starting with 1. 13. 14. 15. 16. 17. 18. Module 5 140 Lesson 1 name _____ date _____ Class_____ Original content Copyright by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Holt McDougal Algebra 2 Practice Introduction to Sequences Find the first 5 terms of each sequence. 1. a1 = 1, an = 3 (an 1) 2. a1 = 2, an = 2 (an 1 + 1) 5 3. a1 = 2, an = (an 1)2 1 _____ _____ _____ 4. a1 = 1, an = 6 2(an 1) 5. a1 = 1, an = (an 1 1)2 3 6. 1122,2nnaaa = = _____ _____ _____ 7. an = (n 2)(n + 1) 8. an = n(2n 1) 9.
10 An = n3 n2 _____ _____ _____ 10. 312nna = 11. an = ( 2)n 1 12. an = n2 2n _____ _____ _____ Write a possible explicit rule for the nth term of each sequence. 13. 8, 16, 24, 32, 40, .. 14. , , , , , .. 15. 3, 6, 11, 18, 27, .. _____ _____ _____ 16..3333 3,,, , ,2 4 8 16 32 17. 2, 1, 4, 7, 10, .. 18. 5, 1, , , , .. _____ _____ _____ Solve. 19. Find the number of line segments in the next two iterations. _____ 20. Jim charges $50 per week for lawn mowing and weeding services. He plans to increase his prices by 4% each year. a. Graph the sequence. b. Describe the pattern. _____ c. To the nearest dollar, how much will he charge per week in 5 years? _____ 594/21/11 4:53:51 PM Houghton Mifflin Harcourt Publishing Company5-1 name Class date Additional PracticeModule 5 141 Lesson 1 name _____ date _____ Class_____ Original content Copyright by Holt McDougal.
