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Section 8.5 Using Recursive Rules with Sequences 441
Essential QuestionEssential Question How can you define a sequence recursively?
A recursive rule gives the beginning term(s) of a sequence and a recursive equation that tells how an is related to one or more preceding terms.
Evaluating a Recursive Rule
Work with a partner. Use each recursive rule and a spreadsheet to write the first six terms of the sequence. Classify the sequence as arithmetic, geometric, or neither. Explain your reasoning. (The figure shows a partially completed spreadsheet for part (a).) a. a 1 = 7, an = an − 1 + 3 b. a 1 = 5, an = an − 1 − 2 c. a 1 = 1, an = 2 an − 1 d. a 1 = 1, an = 1 — 2 ( an − 1 )^2 e. a 1 = 3, an = an − 1 + 1 f. a 1 = 4, an = 1 — 2 an − 1 − 1 g. a 1 = 4, an = 1 — 2 an − 1 h. a 1 = 4, a 2 = 5, an = an − 1 + an − 2
Writing a Recursive Rule
Work with a partner. Write a recursive rule for the sequence. Explain your reasoning. a. 3, 6, 9, 12, 15, 18,... b. 18, 14, 10, 6, 2, −2,... c. 3, 6, 12, 24, 48, 96,... d. 128, 64, 32, 16, 8, 4,... e. 5, 5, 5, 5, 5, 5,... f. 1, 1, 2, 3, 5, 8,...
Writing a Recursive Rule
Work with a partner. Write a recursive rule for the sequence whose graph is shown.
a.
7 − 1
− 1
9 b.
7 − 1
− 1
9
Communicate Your AnswerCommunicate Your Answer
4. How can you define a sequence recursively? 5. Write a recursive rule that is different from those in Explorations 1–3. Write the fi rst six terms of the sequence. Then graph the sequence and classify it as arithmetic, geometric, or neither.
ATTENDING TO
PRECISION
To be proficient in math, you need to communicate precisely to others.
Using Recursive Rules with
Sequences
A B
nth Term 1
n 7 2 10 3 4 5 6
B2+ 3
442 Chapter 8 Sequences and Series
What You Will LearnWhat You Will Learn
Evaluate recursive rules for sequences. Write recursive rules for sequences. Translate between recursive and explicit rules for sequences. Use recursive rules to solve real-life problems.
Evaluating Recursive Rules So far in this chapter, you have worked with explicit rules for the n th term of a sequence, such as an = 3 n − 2 and an = 7(0.5) n. An explicit rule gives a (^) n as a function of the term’s position number n in the sequence. In this section, you will learn another way to define a sequence —by a recursive rule. A recursive rule gives the beginning term(s) of a sequence and a recursive equation that tells how an is related to one or more preceding terms.
Evaluating Recursive Rules
Write the first six terms of each sequence. a. a 0 = 1, an = an − 1 + 4 b. f (1) = 1, f ( n ) = (^3) ⋅ f ( n − 1)
SOLUTION
a. a 0 = 1 a 1 = a 0 + 4 = 1 + 4 = 5 a 2 = a 1 + 4 = 5 + 4 = 9 a 3 = a 2 + 4 = 9 + 4 = 13 a 4 = a 3 + 4 = 13 + 4 = 17 a 5 = a 4 + 4 = 17 + 4 = 21
1st term 2nd term 3rd term 4th term 5th term 6th term
b. f (1) = 1 f (2) = (^3) ⋅ f (1) = 3(1) = 3 f (3) = (^3) ⋅ f (2) = 3(3) = 9 f (4) = (^3) ⋅ f (3) = 3(9) = 27 f (5) = (^3) ⋅ f (4) = 3(27) = 81 f (6) = (^3) ⋅ f (5) = 3(81) = 243
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Write the first six terms of the sequence.
1. a 1 = 3, an = an − 1 − 7 2. a 0 = 162, an = 0.5 a (^) n − 1 3. f (0) = 1, f ( n ) = f ( n − 1) + n 4. a 1 = 4, an = 2 an − 1 − 1
Writing Recursive Rules In part (a) of Example 1, the differences of consecutive terms of the sequence are constant, so the sequence is arithmetic. In part (b), the ratios of consecutive terms are constant, so the sequence is geometric. In general, rules for arithmetic and geometric sequences can be written recursively as follows.
8.5 Lesson
explicit rule, p. 442 recursive rule, p. 442
Core VocabularyCore Vocabullarry
CoreCore ConceptConcept
Recursive Equations for Arithmetic and Geometric Sequences
Arithmetic Sequence an = an − 1 + d, where d is the common difference Geometric Sequence an = r (^) ⋅ an − 1 , where r is the common ratio
444 Chapter 8 Sequences and Series
Translating Between Recursive and Explicit Rules
Translating from Explicit Rules to Recursive Rules
Write a recursive rule for (a) a (^) n = − 6 + 8 n and (b) an = −3( 1 — 2 )
n − 1 .
SOLUTION
a. The explicit rule represents an arithmetic sequence with first term a 1 = − 6 + 8(1) = 2 and common difference d = 8. an = an − 1 + d Recursive equation for arithmetic sequence an = an − 1 + 8 Substitute 8 for d.
A recursive rule for the sequence is a 1 = 2, an = an − 1 + 8.
b. The explicit rule represents a geometric sequence with first term a 1 = −3( —^12 ) 0 = − 3 and common ratio r = (^) —^12. an = r (^) ⋅ an − 1 Recursive equation for geometric sequence an = (^1) — 2 an − 1 Substitute (^1) — 2 for r.
A recursive rule for the sequence is a 1 = −3, an = (^) —^12 an − 1.
Translating from Recursive Rules to Explicit Rules
Write an explicit rule for each sequence. a. a 1 = −5, an = an − 1 − 2 b. a 1 = 10, an = 2 an − 1
SOLUTION
a. The recursive rule represents an arithmetic sequence with first term a 1 = −5 and common difference d = −2. an = a 1 + ( n − 1) d Explicit rule for arithmetic sequence an = − 5 + ( n − 1)(−2) Substitute −5 for a 1 and −2 for d. an = − 3 − 2 n Simplify.
An explicit rule for the sequence is a (^) n = − 3 − 2 n. b. The recursive rule represents a geometric sequence with first term a 1 = 10 and common ratio r = 2. an = a 1 r n^ −^1 Explicit rule for geometric sequence an = 10(2) n^ −^1 Substitute 10 for a 1 and 2 for r.
An explicit rule for the sequence is a (^) n = 10(2) n^ −^1.
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Write a recursive rule for the sequence.
9. an = 17 − 4 n 10. an = 16(3) n^ −^1 Write an explicit rule for the sequence. 11. a 1 = −12, an = an − 1 + 16 12. a 1 = 2, an = − 6 an − 1
Section 8.5 Using Recursive Rules with Sequences 445
Solving Real-Life Problems
Solving a Real-Life Problem
A lake initially contains 5200 fish. Each year, the population declines 30% due to fishing and other causes, so the lake is restocked with 400 fish. a. Write a recursive rule for the number a (^) n of fi sh at the start of the n th year. b. Find the number of fish at the start of the fifth year. c. Describe what happens to the population of fi sh over time.
SOLUTION
a. Write a recursive rule. The initial value is 5200. Because the population declines 30% each year, 70% of the fish remain in the lake from one year to the next. Also, 400 fish are added each year. Here is a verbal model for the recursive equation.
Fish at start of year n
Fish at start of year n − 1
New fi sh added
= 0.7 (^) ⋅ +
an = 0.7 (^) ⋅ an − 1 + 400
A recursive rule is a 1 = 5200, an = (0.7) an − 1 + 400.
b. Find the number of fish at the start of the fifth year. Enter 5200 (the value of a 1 ) in a graphing calculator. Then enter the rule .7 × Ans + 400 to fi nd a 2. Press the enter button three more times to find a 5 ≈ 2262.
There are about 2262 fish in the lake at the start of the fi fth year.
c. Describe what happens to the population of fish over time. Continue pressing enter on the calculator. The screen at the right shows the fish populations for years 44 to 50. Observe that the population of fish approaches 1333.
Over time, the population of fish in the lake stabilizes at about 1333 fish.
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13. WHAT IF? In Example 6, suppose 75% of the fish remain each year. What happens to the population of fish over time?
Check
Set a graphing calculator to sequence and dot modes. Graph the sequence and use the trace feature. From the graph, it appears the sequence approaches 1333.
u=.7*u( -1)+
= X=75 Y=1333.
n
n
5200 .7*Ans+
5200 4040 3228
Section 8.5 Using Recursive Rules with Sequences 447
8.5 Exercises Dynamic Solutions available at BigIdeasMath.com
In Exercises 3–10, write the first six terms of the sequence. (See Example 1.)
3. a 1 = 1 4. a 1 = 1 a (^) n = an − 1 + 3 a (^) n = an − 1 − 5 5. f (0) = 4 6. f (0) = 10 f ( n ) = 2 f ( n − 1) f ( n ) = (^1) — 2 f ( n − 1) 7. a 1 = 2 8. a 1 = 1 an = ( an − 1 )^2 + 1 an = ( an − 1 )^2 − 10 9. f (0) = 2, f (1) = 4 f ( n ) = f ( n − 1) − f ( n − 2) 10. f (1) = 2, f (2) = 3
f ( n ) = f ( n − 1) ⋅ f ( n − 2)
In Exercises 11–22, write a recursive rule for the sequence. (See Examples 2 and 3.)
11. 21, 14, 7, 0, −7,... 12. 54, 43, 32, 21, 10,... 13. 3, 12, 48, 192, 768,... 14. 4, −12, 36, −108,... 15. 44, 11, (^11) — 4 , (^11) — 16 , (^) —^1164 ,... 16. 1, 8, 15, 22, 29,... 17. 2, 5, 10, 50, 500,... 18. 3, 5, 15, 75, 1125,... 19. 1, 4, 5, 9, 14,... 20. 16, 9, 7, 2, 5,... 21. 6, 12, 36, 144, 720,... 22. −3, −1, 2, 6, 11,...
In Exercises 23 –26, write a recursive rule for the sequence shown in the graph.
23.
n
f ( n ) 4
2
2 4
n
8 f ( n )
4
2 4
n
f ( n ) 4
2 4
n
f ( n ) 4
2
ERROR ANALYSIS In Exercises 27 and 28, describe and correct the error in writing a recursive rule for the sequence 5, 2, 3, − 1, 4,.. ..
27. Beginning with the third term in the sequence, each term a (^) n equals an − 2 − an − 1. So, a recursive rule is given by an = an − 2 − an − 1.
Beginning with the second term in the sequence, each term a (^) n equals an − 1 − 3. So, a recursive rule is given by a 1 = 5, an = an − 1 − 3.
In Exercises 29–38, write a recursive rule for the sequence. (See Example 4.)
29. an = 3 + 4 n 30. an = − 2 − 8 n 31. an = 12 − 10 n 32. a (^) n = 9 − 5 n 33. an = 12(11) n^ −^1 34. an = −7(6) n^ −^1 35. an = 2.5 − 0.6 n 36. an = −1.4 + 0.5 n 37. an = −^1 —
—
n − 1
38. an = (^1) — 4
(5) n^ −^1
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
1. COMPLETE THE SENTENCE A recursive _________ tells how the n th term of a sequence is related to one or more preceding terms. 2. WRITING Explain the difference between an explicit rule and a recursive rule for a sequence.
Vocabulary and Core Concept CheckVocabulary and Core Concept Check
448 Chapter 8 Sequences and Series
39. REWRITING A FORMULA
You have saved $82 to buy a bicycle. You save an additional $30 each month. The explicit rule an = 30 n + 82 gives the amount saved after n months. Write a recursive rule for the amount you have saved n months from now.
40. REWRITING A FORMULA Your salary is given by the explicit rule a (^) n = 35,000(1.04) n^ −^1 , where n is the number of years you have worked. Write a recursive rule for your salary.
In Exercises 41–48, write an explicit rule for the sequence. (See Example 5.)
41. a 1 = 3, an = an − 1 − 6 42. a 1 = 16, an = an − 1 + 7 43. a 1 = −2, an = 3 an − 1 44. a 1 = 13, an = 4 an − 1 45. a 1 = −12, an = an − 1 + 9. 46. a 1 = −4, an = 0.65 an − 1 47. a 1 = 5, an = an − 1 − 1 — 3 48. a 1 = −5, an = 1 — 4 an − 1 49. REWRITING A FORMULA A grocery store arranges cans in a pyramid-shaped display with 20 cans in the bottom row and two fewer cans in each subsequent row going up. The number of cans in each row is represented by the recursive rule a 1 = 20, an = an − 1 − 2. Write an explicit rule for the number of cans in row n. 50. REWRITING A FORMULA The value of a car is given by the recursive rule a 1 = 25,600, an = 0.86 an − 1 , where n is the number of years since the car was new. Write an explicit rule for the value of the car after n years. 51. USING STRUCTURE What is the 1000th term of the sequence whose first term is a 1 = 4 and whose n th term is a (^) n = an − 1 + 6? Justify your answer.
A^4006 B^5998
C^1010 D 10,
52. USING STRUCTURE What is the 873rd term of the sequence whose first term is a 1 = 0.01 and whose n th term is a (^) n = 1.01 an − 1? Justify your answer.
A 58.65^ B 8.
C 1.08^ D 586,459.
53. PROBLEM SOLVING An online music service initially has 50,000 members. Each year, the company loses 20% of its current members and gains 5000 new members. (See Example 6.)
Key:
Beginning of first year
50, members
Beginning of second year
20% leave
5000 45,000 members join
=
= 5000 members = join = leave
a. Write a recursive rule for the number a (^) n of members at the start of the n th year. b. Find the number of members at the start of the fifth year. c. Describe what happens to the number of members over time.
54. PROBLEM SOLVING You add chlorine to a swimming pool. You add 34 ounces of chlorine the first week and 16 ounces every week thereafter. Each week, 40% of the chlorine in the pool evaporates.
First week Each successive week
16 oz of chlorine are added 40% of chlorine has evaporated
34 oz of chlorine are added
a. Write a recursive rule for the amount of chlorine in the pool at the start of the n th week. b. Find the amount of chlorine in the pool at the start of the third week. c. Describe what happens to the amount of chlorine in the pool over time.
55. OPEN-ENDED Give an example of a real-life situation which you can represent with a recursive rule that does not approach a limit. Write a recursive rule that represents the situation. 56. OPEN-ENDED Give an example of a sequence in which each term after the third term is a function of the three terms preceding it. Write a recursive rule for the sequence and find its first eight terms.
450 Chapter 8 Sequences and Series
64. THOUGHT PROVOKING Let a 1 = 34. Then write the terms of the sequence until you discover a pattern.
an + 1 =
(^1) — 2 an ,^ if^ an^ is even 3 a (^) n + 1, if an is odd Do the same for a 1 = 25. What can you conclude?
65. MODELING WITH MATHEMATICS You make a $500 down payment on a $3500 diamond ring. You borrow the remaining balance at 10% annual interest compounded monthly. The monthly payment is $213.59. How long does it take to pay back the loan? What is the amount of the last payment? Justify your answers. 66. HOW DO YOU SEE IT? The graph shows the first six terms of the sequence a 1 = p , an = ran − 1.
n
an (1, p )
a. Describe what happens to the values in the sequence as n increases. b. Describe the set of possible values for r. Explain your reasoning.
67. REASONING The rule for a recursive sequence is as follows. f (1) = 3, f (2) = 10 f ( n ) = 4 + 2 f ( n − 1) − f ( n − 2) a. Write the first fi ve terms of the sequence. b. Use finite differences to find a pattern. What type of relationship do the terms of the sequence show? c. Write an explicit rule for the sequence. 68. MAKING AN ARGUMENT Your friend says it is impossible to write a recursive rule for a sequence that is neither arithmetic nor geometric. Is your friend correct? Justify your answer. 69. CRITICAL THINKING The first four triangular numbers Tn and the fi rst four square numbers Sn are represented by the points in each diagram.
1
1
2 3 4
2 3 4
a. Write an explicit rule for each sequence. b. Write a recursive rule for each sequence. c. Write a rule for the square numbers in terms of the triangular numbers. Draw diagrams to explain why this rule is true.
70. CRITICAL THINKING You are saving money for retirement. You plan to withdraw $30,000 at the beginning of each year for 20 years after you retire. Based on the type of investment you are making, you can expect to earn an annual return of 8% on your savings after you retire. a. Let an be your balance n years after retiring. Write a recursive equation that shows how a (^) n is related to an − 1_._ b. Solve the equation from part (a) for an − 1. Find a 0 , the minimum amount of money you should have in your account when you retire. ( Hint: Let a 20 = 0.)
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Solve the equation. Check your solution. (Section 5.4)
71. √— x + 2 = 7 72. 2 √ — x − 5 = 15 73. √^3 — x + 16 = 19 74. 2 √^3 — x − 13 = − 5
The variables x and y vary inversely. Use the given values to write an equation relating x and y****. Then find y when x = 4. (Section 7.1)
75. x = 2, y = 9 76. x = −4, y = 3 77. x = 10, y = 32
Reviewing what you learned in previous grades and lessons
