How do arithmetic sequences work?

Generalizing Arithmetic Sequences

Lesson Objective: Students will learn the vocabulary and notation for arithmetic sequences as they develop formulas for the nth

term.

Length of Activity: One day (approximately 50 minutes)

Core Problems: Problems 2-61 and 2-62

Ways of Thinking: Choosing a strategy, generalizing

Materials: None

Lesson Overview: In this lesson, students’ work will center on the relationship between arithmetic sequences and linear functions. Students may have drawn this connection already, but if not, this lesson highlights the similarities and differences and provides opportunities to recall and sharpen

students’ skills in working with linear equations, graphs, tables, and applications. In order to make these connections stronger, students are

2 2 2 2 2

expected to consider the first term of a sequence the “initial” term, with n=0. In Lesson 2.1.8, students will transition to the more traditional approach, in which the first term is considered to be n=1. This lesson consists of more problems than should be assigned in one class period. Problems 2-61 and 2-62 are essential because they

introduce the language and notation that students will need. The rest of the problems provide opportunities for practice and application. Some of them are quite challenging. For average classes, it would be

appropriate to skip problems 2-67 through 2-69. Be sure that you have worked the problems you assign before class.

Suggested Lesson Activity:

When you introduce today’s lesson, focus on the fact that the students’ task is to identify and share strategies for finding rules for arithmetic sequences. As they work in their study teams, they should both

articulate their own strategies and listen for the strategies that others are using. As you observe teams choosing different strategies, you may decide to interrupt their work to ask students to present to the entire class, or you may leave this until the end of the day as closure. Problems 2-61 and 2-62 review and introduce the vocabulary and notation for arithmetic sequences. They also connect arithmetic sequences to linear functions while clarifying the differences in their domains. Do not worry if students are not completely clear on this yet, as it will be the central focus of Lesson 2.1.7.

The remaining problems in this lesson alternate between context-based problems and those that are more abstract. Students are given

information and need to choose appropriate strategies. For example, problems 2-63 and 2-64 both ask for a rule for an arithmetic sequence given two nonconsecutive terms. Teams can use a variety of methods to solve the problem, but if they are stuck, consider asking questions such as, “Can a different representation help you get more information?” or “How could you use your knowledge of linear functions?” Students may use algebraic strategies or logical reasoning to find rules. Do not expect any particular method at this point. Instead, help students to be careful about notation and language, such as using n and t(n).

Suggested Study Team Collaboration Strategy: for problem 2-61, Reciprocal Teaching

Closure: (10 minutes)

Bring the class together and have teams share strategies for finding rules for arithmetic sequences based on multiple representations. Consider asking questions such as: “How could you use a table to find the rule for an arithmetic sequence?” “Did any team use an equation? How?” “How could you use a graph?”

Homework: Problems 2-71 through 2-77

Note: Problem 2-74 is intended to prepare students for Lesson 2.1.6, so be sure to assign it.

2.1.5 How do arithmetic sequences work?

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Generalizing Arithmetic Sequences

Student pages for this lesson are 75 – 78.

In Lesson 2.1.4, you learned how to identify arithmetic and geometric sequences. Today you will solve problems involving arithmetic sequences. Use the questions below to help your team stay focused and start mathematical conversations.

What type of sequence is this? How do we know? How can we find the rule?

Is there another way to see it?

2-61. LEARNING THE LANGUAGE OF SEQUENCES

Sequences have their own notation and special words and phrases that help describe them, such as “term” and “term number.” The questions below will help you learn more of this vocabulary and notation.

Consider the sequence –9, –5, –1, 3, 7, … as you complete parts (a) through (f) below.

[ a: It is arithmetic, since its generator is +4; b: –9, add 4; c: The difference is 4, the same as the generator; d: t(n)=4n!9; e: It should be discrete, since the rule is for a sequence (see graph below right); f: It is the slope, since it

represents the growth per term. ]

a. Is this sequence arithmetic, geometric, or neither? How can you tell? b. What are the initial value and the generator for the sequence?

c. What is the difference between each term and the term before it? How is this related to the generator? For an arithmetic sequence, this is also known as the

common difference.

d. Find a rule (beginning t(n)= ) for the nth

term of this sequence. You can assume that for the first term of the sequence, n=0.

e. Graph your rule. Should the graph be continuous or discrete? Why?

f. How is the common difference related to the graph of your rule? Why does y

2-62. Consider the sequence t(n) = –4, –1, 2, 5, … [ a: t(n)=3n!4; b: No, because the domain of t(n) includes only positive integers, none of which gives an output value of 42; c: Yes, f(x)can be equal to 42 (when x= 46₃ ),because it has a domain of all real number; d: Part (a) has a domain limited to integers, while the domain of part (b) is all real numbers including non-integers. ]

a. Write a rule for t(n).

b. Is it possible for t(n) to equal 42? Justify your answer.

c. For the function f(x)=3x!4, is it possible for f(x) to equal 42? Explain. d. Explain the difference between t(n) and f(x) that makes your answers to parts

(b) and (c) different.

2-63. Trixie wants to create an especially tricky arithmetic sequence. She wants the 5th term of the sequence to equal 11 and the 50th

term to equal 371. That is, she wants

t(4)=11 and t(49)=371. Is it possible to create an arithmetic sequence to fit her information? If it is possible, find the rule, the initial value t(0), and the common difference for the arithmetic sequence. If it is not possible, explain why not.

[ t(n)=8n!21, t(0)=–21, common difference = 8 ]

2-64. Seven years ago, Kodi found a box of old baseball cards in the garage. Since then, he has added a consistent number of cards to the collection each year. He had 52 cards in the collection after 3 years and now has 108 cards. [ a: 10 cards, b: 248 cards, c: t(n)=14n+10 ]

a. How many cards were in the original box?

b. Kodi plans to keep the collection for a long time. How many cards will the collection contain 10 years from now?

c. Write a rule that determines the number of cards in the collection after n years. What does each number in your rule represent?

2-65. Trixie now wants an arithmetic sequence with a common difference of –17 and a 16th

term of 93. (In other words, t(15)=93.) Is it possible to create an arithmetic sequence to fit her information? If it is possible, find the rule. If it is not possible, explain why not. [ t(n)=!17n+348 ]

2-66. Your favorite radio station, WCPM, is having a contest. The DJ poses a question to the listeners. If the caller answers correctly, he or she wins the prize money. If the caller answers incorrectly, $20 is added to the prize money and the next caller is eligible to win. The current question is difficult, and no one has won for two days.

[ a: $455; b: $1360=$20n+$100, so 63 people would have to guess incorrectly. ]

a. Lucky you! Fourteen people already called in today with incorrect answers, so when you called (with the right answer, of course) you won $735! How much was the prize worth at the beginning of the day today?

b. Suppose the contest always starts with $100. How many people would have to guess incorrectly for the winner to get $1360?

2-67. Trixie is at it again. This time she wants an arithmetic sequence that has a graph with a slope of 22. She also wants t(8)=164 and the 13th

term to have a value of 300. Is it possible to create an arithmetic sequence to fit her information? If it is possible, find the rule. If it is not possible, explain why not. [ It is not possible. The graph of rule for the two given terms would have a slope of 34, not 22. ]

2-68. Find the rule for each arithmetic sequence represented by the n!t(n) tables below. [ a: t(n)=11n!23, b: t(n)=!9 8n+122.5 ] a. n t(n) b. 7 54 3 10 19 186 16 153 40 417 n t(n) 100 10 20 100

2-69. Trixie decided to extend her trickiness to tables. Each n!t(n) table below

represents an arithmetic sequence. Find expressions for the missing terms and write a rule. [ a: Since the common difference is 7!t(0), the 2nd

term is 7+(7!t(0))

and the 3rd

term is 7+(7!t(0))+(7!t(0)), and the rule is

t(n)=(7!t(0))"n+t(0); b: The common difference is f ! p, the initial value is p!(f ! p), and the rule is t(n)=(f ! p)"n+ p!(f ! p). ]

a. b.

2-70. Trixie exclaimed, “Hey! Arithmetic sequences are just another name for linear functions.” What do you think? Justify your idea based on multiple representations.

[ Answers vary. ]

2-71. Determine whether 447 is a term of each sequence below. If so, which term is it?

[ a: Yes, the 91st

term or t(90)=447; b: no; c: Yes, the 153rd

term or t(152)=447; d: no; e: No, n=!64 is not in the domain. ]

a. t(n)=5n– 3 b. t(n)=24 – 5n

c. t(n)=–6+3(n– 1) d. t(n)=14 – 3n

e. t(n)=!8!7(n!1)

2-72. Choose one of the sequences in problem 2-71 for which you determined that 447 is not a term. Write a clear explanation (that an Algebra 1 student would be able to understand) describing how you can be sure that 447 is not a term of the sequence.

[ Justifications vary. ]

2-73. Find the common difference for each sequence listed below. Write an expression for the nth

term in each sequence below, keeping in mind that the first term of each sequence is t(0). [ a: m=3, t(n)=3n+4; b: m=5, t(n)=5n+3; c: m=!5, t(n)= –5n+24; d: m=2.5, t(n)=2.5n+7 ] a. 4, 7, 10, 13, … b. 3, 8, 13, … n t(n) 0 t(0) 1 7 2 3 4 n t(n) 0 1 p 2 f

2-74. Great Amusements Park has been raising its ticket prices every year, as shown in the table at right. [ a: $88.58; b: Descriptions vary, but students may say they are multiplying by 1.1 or growing by 10% each year. ]

a. What will the price of admission be in year 6? b. Describe how the ticket prices are growing.

2-75. Solve the system at right for m and b. [ m=13, b=17 ]

2-76. Multiply each expression below. [ a: x2!_5x!_{14, b: 6m}2+_11m!_7,

c: x2 !_6x+_{9, d: 4y}2 !_{9 ]}

a. (x+2)(x!7) b. (3m+7)(2m!1)

c. (x!3)2 d. (2y+3)(2y!3)

2-77. Simplify each expression. [ a: 9x2_{, b: 18x}2_{, c:} 6

x, d: 32x ] a. (3x)2 b. 2(3x)2 c. 2(3x)2 3x3 d. 2(3x)2 (3x)3 Year Price 0 $50 1 $55 2 $60.50 3 $66.55 1239=94m+b !810=61m+b