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Section 4.6 Arithmetic Sequences 209
4.6 Arithmetic Sequences
Describing a Pattern
Work with a partner. Use the fi gures to complete the table. Plot the points given by your completed table. Describe the pattern of the y -values. a. n = 1 n = 2 n = 3 n = 4 n = 5
Number of stars, n^1 2 3 4 Number of sides, y
b. n = 1 n = 2 n = 3 n = 4 n = 5
n^1 2 3 4 Number of circles, y
c. n = 1 n = 2 n = 3 n = 4 n = 5
Number of rows, n 1 2 3 4 5 Number of dots, y
Communicate Your AnswerCommunicate Your Answer
2. How can you use an arithmetic sequence to describe a pattern? Give an example from real life. 3. In chemistry, water is called H 2 O because each molecule of water has two hydrogen atoms and one oxygen atom. Describe the pattern shown below. Use the pattern to determine the number of atoms in 23 molecules. n = 1 n = 2 n = 3 n = 4 n = 5
Essential QuestionEssential Question How can you use an arithmetic sequence to
describe a pattern?
An arithmetic sequence is an ordered list of numbers in which the difference between each pair of consecutive terms , or numbers in the list, is the same.
LOOKING FOR
A PATTERN
To be proficient in math, you need to look closely to discern patterns and structure.
n
y
0 1 2 3 4 5 0
10
20
30
40
50
60
n
y
0 1 2 3 4 5 0
1
2
3
4
5
6
n
y
(^001 2 3 4 )
2
4
6
8
10
12
210 Chapter 4 Writing Linear Functions
4.6 Lesson^ What You Will LearnWhat You Will Learn
Write the terms of arithmetic sequences. Graph arithmetic sequences. Write arithmetic sequences as functions.
Writing the Terms of Arithmetic Sequences A sequence is an ordered list of numbers. Each number in a sequence is called a term. Each term an has a specific position n in the sequence. 5, 10, 15, 20, 25,... , an ,...
sequence, p. 210 term, p. 210 arithmetic sequence, p. 210 common difference, p. 210 Previous point-slope form function notation
Core VocabularyCore Vocabullarry
Extending an Arithmetic Sequence
Write the next three terms of the arithmetic sequence. −7, −14, −21, −28,...
SOLUTION
Use a table to organize the terms and find the pattern.
Add −7 to a term to find the next term.
The next three terms are −35, −42, and −49.
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Write the next three terms of the arithmetic sequence.
READING An ellipsis (.. .) is a series of dots that indicates an intentional omission of information. In mathematics, the... notation means “and so forth.” The ellipsis indicates that there are more terms in the sequence that are not shown.
CoreCore ConceptConcept
Arithmetic Sequence
In an arithmetic sequence , the difference between each pair of consecutive terms is the same. This difference is called the common difference. Each term is found by adding the common difference to the previous term. 5, 10, 15, 20,... Terms of an arithmetic sequence
- 5 + 5 + 5 common difference
1st position 3rd position n th position
Each term is 7 less than the previous term. So, the common difference is −7.
Position 1 2 3 4 Term − 7 − 14 − 21 − 28
+(−7) +(−7) +(−7)
Position 1 2 3 4 5 6 7 Term − 7 − 14 − 21 − 28 − 35 − 42 − 49
+(−7) +(−7) +(−7)
212 Chapter 4 Writing Linear Functions
Writing Arithmetic Sequences as Functions Because consecutive terms of an arithmetic sequence have a common difference, the sequence has a constant rate of change. So, the points represented by any arithmetic sequence lie on a line. You can use the first term and the common difference to write a linear function that describes an arithmetic sequence. Let a 1 = 4 and d = 3. Position, n Term, an Written using a 1 and d Numbers 1 fi rst term, a 1 a 1 4 2 second term, a 2 a 1 + d 4 + 3 = 7 3 third term, a 3 a 1 + 2 d 4 + 2(3) = 10 4 fourth term, a 4 a 1 + 3 d 4 + 3(3) = 13 … … … …
n n th term, an a 1 + ( n − 1) d 4 + ( n − 1)(3)
CoreCore ConceptConcept
Equation for an Arithmetic Sequence
Let an be the n th term of an arithmetic sequence with first term a 1 and common difference d. The n th term is given by an = a 1 + ( n − 1) d.
Finding the n th Term of an Arithmetic Sequence
Write an equation for the n th term of the arithmetic sequence 14, 11, 8, 5,.. .. Then fi nd a 50.
SOLUTION
The fi rst term is 14, and the common difference is −3. an = a 1 + ( n − 1) d Equation for an arithmetic sequence an = 14 + ( n − 1)(−3) Substitute 14 for a 1 and −3 for d. an = − 3 n + 17 Simplify. Use the equation to find the 50th term. an = − 3 n + 17 Write the equation. a 50 = −3(50) + 17 Substitute 50 for n. = − 133 Simplify.
The 50th term of the arithmetic sequence is −133.
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Write an equation for the n th term of the arithmetic sequence. Then find a 25.
8. 4, 5, 6, 7,... 9. 8, 16, 24, 32,...
ANOTHER WAY
An arithmetic sequence is a linear function whose domain is the set of positive integers. You can think of d as the slope and (1, a 1 ) as a point on the graph of the function. An equation in point-slope form for the function is a (^) n − a 1 = d ( n − 1). This equation can be rewritten as an = a 1 + ( n − 1) d.
STUDY TIP
Notice that the equation in Example 4 is of the form y = mx + b , where y is replaced by an and x is replaced by n.
Section 4.6 Arithmetic Sequences 213
You can rewrite the equation for an arithmetic sequence with first term a 1 and common difference d in function notation by replacing a (^) n with f ( n ). f ( n ) = a 1 + ( n − 1) d The domain of the function is the set of positive integers.
Writing Real-Life Functions
Online bidding for a purse increases by $5 for each bid after the $60 initial bid.
a. Write a function that represents the arithmetic sequence. b. Graph the function. c. The winning bid is $105. How many bids were there?
SOLUTION
a. The first term is 60, and the common difference is 5. f ( n ) = a 1 + ( n − 1) d Function for an arithmetic sequence f ( n ) = 60 + ( n − 1)5 Substitute 60 for a 1 and 5 for d. f ( n ) = 5 n + 55 Simplify.
The function f ( n ) = 5 n + 55 represents the arithmetic sequence. b. Make a table. Then plot the ordered pairs ( n , an ).
Bid number, n
Bid amount, a (^) n 1 60 2 65 3 70 4 75
0
Bid amount (dollars) 0 n
a (^) n
Bid number
Bidding on a Purse
1 2 3 4 5 6
55
60
65
70
75
80
(1, 60)
(2, 65)
(3, 70)
(4, 75)
c. Use the function to find the value of n for which f ( n ) = 105. f ( n ) = 5 n + 55 Write the function. 105 = 5 n + 55 Substitute 105 for f ( n ). 10 = n Solve for n.
There were 10 bids.
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11. A carnival charges $2 for each game after you pay a $5 entry fee.
a. Write a function that represents the arithmetic sequence.
b. Graph the function.
c. How many games can you play when you take $29 to the carnival?
Games Total cost 1 $ 2 $ 3 $ 4 $
REMEMBER
The domain is the set of positive integers.
Bid number^1 2 3 Bid amount $60^ $65^ $70^ $
Section 4.6 Arithmetic Sequences 215
32. FINDING A PATTERN Write a sequence that represents the sum of the numbers in each roll. Is the sequence arithmetic? Explain.
Roll 1 Roll 2 Roll 3 Roll 4
In Exercises 33−38, write an equation for the n th term of the arithmetic sequence. Then find a 10. (See Example 4.)
33. −5, −4, −3, −2,... 34. −6, −9, −12, −15,... 35.^1 — 2 , 1, 1 1 — 2 , 2,... 36. 100, 110, 120, 130,... 37. 10, 0, −10, −20,... 38.^3 — 7 , 4 — 7 , —^57 , —^67 ,... 39. ERROR ANALYSIS Describe and correct the error in finding the common difference of the arithmetic sequence.
The common difference is 1.
40. ERROR ANALYSIS Describe and correct the error in writing an equation for the n th term of the arithmetic sequence.
an = a 1 + nd an = 14 + 8 n
41. NUMBER SENSE The first term of an arithmetic sequence is 3. The common difference of the sequence is 1.5 times the first term. Write the next three terms of the sequence. Then graph the sequence. 42. NUMBER SENSE The first row of a dominoes display has 10 dominoes. Each row after the first has two more dominoes than the row before it. Write the first fi ve terms of the sequence that represents the number of dominoes in each row. Then graph the sequence.
REPEATED REASONING In Exercises 43 and 44, (a) draw the next three figures in the sequence and (b) describe the 20th fi gure in the sequence.
43.
45. MODELING WITH MATHEMATICS The total number of babies born in a country each minute after midnight January 1st can be estimated by the sequence shown in the table. (See Example 5.)
Minutes after midnight January 1st
Total babies born 5 10 15 20
a. Write a function that represents the arithmetic sequence. b. Graph the function. c. Estimate how many minutes after midnight January 1st it takes for 100 babies to be born.
46. MODELING WITH MATHEMATICS The amount of money a movie earns each week after its release can be approximated by the sequence shown in the graph.
0 0
Earnings
(millions of dollars) n
a (^) n
Week
Movie Earnings
1 2 3 4 5
10
20
30
40
50
(^60) (1, 56) (2, 48) (3, 40) (4, 32)
a. Write a function that represents the arithmetic sequence. b. In what week does the movie earn $16 million? c. How much money does the movie earn overall?
216 Chapter 4 Writing Linear Functions
Maintaining Mathematical ProficiencyMaintaining Mathematical Proficiency
Solve the inequality. Graph the solution. (Section 2.2)
58. x + 8 ≥ − 9 59. 15 < b − 4 60. t − 21 < − 12 61. 7 + y ≤ 3
Graph the function. Compare the graph to the graph of f ( x ) = ∣ x ∣. Describe the domain and range. (Section 3.7)
62. h ( x ) = 3 ∣ x ∣ 63. v ( x ) = ∣ x − 5 ∣ 64. g( x ) = ∣ x ∣ + 1 65. r ( x ) = − 2 ∣ x ∣
Reviewing what you learned in previous grades and lessons
MATHEMATICAL CONNECTIONS In Exercises 47 and 48, each small square represents 1 square inch. Determine whether the areas of the figures form an arithmetic sequence. If so, write a function f that represents the arithmetic sequence and find f (30).
47.
49. REASONING Is the domain of an arithmetic sequence discrete or continuous? Is the range of an arithmetic sequence discrete or continuous? 50. MAKING AN ARGUMENT Your friend says that the range of a function that represents an arithmetic sequence always contains only positive numbers or only negative numbers. Your friend claims this is true because the domain is the set of positive integers and the output values either constantly increase or constantly decrease. Is your friend correct? Explain. 51. OPEN-ENDED Write the first four terms of two different arithmetic sequences with a common difference of −3. Write an equation for the n th term of each sequence. 52. THOUGHT PROVOKING Describe an arithmetic sequence that models the numbers of people in a real-life situation. 53. REPEATED REASONING Firewood is stacked in a pile. The bottom row has 20 logs, and the top row has 14 logs. Each row has one more log than the row above it. How many logs are in the pile? 54. HOW DO YOU SEE IT? The bar graph shows the costs of advertising in a magazine.
0
10,
20,
30,
40,
50,
60,
70,
Cost (dollars)
Size of advertisement (pages)
Magazine Advertisement
1 2 3 4
a. Does the graph represent an arithmetic sequence? Explain. b. Explain how you would estimate the cost of a six-page advertisement in the magazine.
55. REASONING Write a function f 1 4 12
that represents the arithmetic sequence shown in the mapping diagram.
56. PROBLEM SOLVING A train stops at a station every 12 minutes starting at 6:00 a.m. You arrive at the station at 7:29 a.m. How long must you wait for the train? 57. ABSTRACT REASONING Let x be a constant. Determine whether each sequence is an arithmetic sequence. Explain.
a. x + 6, 3 x + 6, 5 x + 6, 7 x + 6,... b. x + 1, 3 x + 1, 9 x + 1, 27 x + 1,...
