LESSON
4.3
Name
Modeling with
Arithmetic Sequences
Class
4.3
Date
Modeling with Arithmetic
Sequences
Essential Question: How can you solve real-world problems using arithmetic sequences?
Texas Math Standards
Resource
Locker
A1.2.A determine the domain and range of a linear function… discrete. Also A1.12.D, A1.12.B
The student is expected to:
Explore
A1.2.A
Interpreting Models of Arithmetic Sequences
Determine the domain and range of a linear function in mathematical
problems; determine reasonable domain and range values for real-world
situations, ... discrete; .... Also A1.12.D, A1.12.B
You can model real-world situations and solve problems using
models of arithmetic sequences. For example, suppose watermelons
cost $6.50 each at the local market. The total cost, in dollars, of
n watermelons can be found using c(n) = 6.5n.
Mathematical Processes

A1.1.A
Complete the table of values for 1, 2, 3, and 4 watermelons.
Watermelons n
Total cost ($) c(n)
Apply mathematics to problems arising in everyday life, society, and the
workplace.

Language Objective
1
2
3
4
6.5
13
19.5
26
What is the common difference?
6.5
4.C
Interpret the meaning of questions about real-world situations.
ENGAGE
Essential Question: How can you solve
real-world problems using arithmetic
sequences?
If a real-world situation can be modeled using an
arithmetic sequence, you can write a rule for the
sequence and use the rule to solve the problem.
PREVIEW: LESSON
PERFORMANCE TASK
View the online Engage. Discuss the photo and that
an ant colony consists of an egg-laying queen, a large
number of male ants, and female worker ants, who
gather food and build, maintain, and guard the nest.
Then preview the Lesson Performance Task.
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©aerogondo2/Shutterstock

What does n represent in this context?
The variable n represents the number of watermelons.

What are the dependent and independent variables in this context?
The variable n is the independent variable and c(n) is the dependent variable.

Find c(7). What does this value represent?
c(7) = 6.5 ⋅ 7 = 45.5
This value represents the cost of 7 watermelons: $45.50.
Reflect
1.
Discussion What domain values make sense for c(n) = 6.5n in this situation?
Only nonnegative integers make sense for the domain of c(n) = 6.5n in this situation.
Module 4
ges
EDIT--Chan
DO NOT Key=TX-B
Correction
must be
Lesson 3
171
gh "File info"
made throu
Date
Class
metic
with Arith
Modeling
Sequences
Name
4.3
ms using
ion: How
Quest
Essential
can you
orld proble
solve real-w
n and
ine the domai
A1.2.A determ
A1_MTXESE346734_U2M04L3.indd 171
ting
Interpre
Explore
arithmetic
Resource
Locker
sequences?
, A1.12.B
e. Also A1.12.D
n… discret
linear functio
range of a
HARDCOVER PAGES 135144
ces
tic Sequen
Arithme
Models of
using
problems
and solve
watermelons
situations
le, suppose
, of
real-world
For examp
cost, in dollars
You can model etic sequences.
t. The total
arithm
local marke
models of
= 6.5n.
each at the
using c(n)
cost $6.50
elons.
be found
4 waterm
elons can
2, 3, and
n waterm
the table
Complete

of values
1
ns n
Watermelo
($) c(n)
Total cost
What is the

6.5
Turn to these pages to
find this lesson in the
hardcover student
edition.
for 1,
2
13
3
19.5
4
26
difference?
common
6.5
What does

Credits:
y • Image
g Compan
What are
t?
ble.
in this contex
t variables
ndent varia
independen
is the depe
dent and
and c(n)
the depen
nt variable
independe
ble n is the
The varia
Publishin

Harcour t
n Mifflin
© Houghto
utterstock
ndo2/Sh
©aerogo
t?
in this contex
melons.
of water
n represent
number
sents the
ble n repre
The varia
represent?
this value
What does
Find c(7).
$45.50.
⋅ 7 = 45.5
melons:
c(7) = 6.5
of 7 water
the cost
represents
This value

on?
tion.
in this situati
this situa
c(n) = 6.5n
= 6.5n in
sense for
in of c(n)
the doma
n values make
sense for
What domai
ers make
Discussion
egative integ
Only nonn
Reflect
1.
Lesson 3
171
Module 4
171
Lesson 4.3
L3.indd
4_U2M04
SE34673
A1_MTXE
171
2/14/15
9:27 AM
2/14/15 9:27 AM
Modeling Arithmetic Sequences From a Table
Explain 1
EXPLORE
Given a table of data values from a real-world situation involving an arithmetic sequence, you can construct a
function model and use it to solve problems.
Example 1

Interpreting Models of Arithmetic
Sequences
Construct an explicit rule in function notation for the arithmetic sequence represented in
the table. Then interpret the meaning of a specific term of the sequence in the given context.
Suppose the table shows the cost, in dollars, of postage per ounce of a letter.
Number of ounces
n
1
2
3
4
Cost ($) of postage
f(n)
0.35
0.55
0.75
0.95
INTEGRATE TECHNOLOGY
Determine the value of ƒ(9), and tell what it represents in this situation.
Students have the option of completing the activity
either in the book or online.
Find the common difference, d. d = 0.55 - 0.35 = 0.20
Substitute 0.35 for ƒ(1) and 0.20 for d.
ƒ(n) = ƒ(1) + d(n - 1)
CONNECT VOCABULARY
ƒ(n) = 0.35 + 0.20(n - 1)
ƒ(9) = 0.35 + 0.20(8) = 1.95
Remind students that a variable is a quantity that can
change or take on different values. An independent
variable is a value that does not depend on any other
variable. A dependent variable depends on the
independent variable.
So, the cost of postage for a 9-ounce letter is $1.95.

The table shows the cumulative total interest paid, in dollars, on a loan after each month.
Number of months
n
1
2
3
4
Cumulative total ($)
f(n)
160
230
300
370
Determine the value of ƒ(20) and tell what it represents in this situation.
Find the common difference, d. d = 230 - 160 = 70
Substitute 160 for ƒ(1) and 70 for d.
EXPLAIN 1
ƒ(n) = 160 + 70 (n - 1)
Modeling Arithmetic Sequences From
a Table
ƒ(n) = ƒ(1) + d(n - 1)
Find ƒ(20) and interpret the value in context.
ƒ(n) = ƒ(1) + d(n - 1)
(
20
)=
160
So, the cumulative total
+
70
(
interest paid after
19
20
)=
1490
months is
$1490 .
Your Turn
Construct an explicit rule in function notation for the arithmetic sequence represented in the table.
Then interpret the meaning of a specific term of the sequence in the given context.
2.
The table shows ƒ(n), the distance, in miles, from the store after Mila has traveled for n minutes.
Time (min)
Distance (mi)
n
1
2
3
4
f(n)
20
32
44
56
© Houghton Mifflin Harcourt Publishing Company
ƒ
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
Review the connections among an arithmetic
sequence, the table representing the sequence, and an
explicit rule for the sequence.
Determine the value of ƒ(10) and tell what it represents in this situation.
d = 12 f(n) = 20 + 12(n - 1)
f(10) = 20 + 12(9) = 128
Module 4
So, Mila is 128 miles from the store after 10 minutes.
172
Lesson 3
PROFESSIONAL DEVELOPMENT
A1_MTXESE346734_U2M04L3 172
Integrate Mathematical Processes
2/22/14 3:44 AM
This lesson provides an opportunity to address Mathematical Process TEKS
A1.1.A, which calls for students to “apply mathematics to problems arising in
everyday life, society, and the workplace.” In this lesson, students interpret models
of arithmetic sequences in everyday situations represented in tables, graphs, and
word descriptions.
Modeling with Arithmetic Sequences
172
3.
EXPLAIN 2
Number of battery packs
Total cost ($)
Modeling Arithmetic Sequences from
a Graph
n
1
2
3
4
f(n)
4.90
8.90
12.90
16.90
Determine the value of ƒ(18) and tell what it represents in this situation.
d = 4 f(n) = 4.90 + 4(n - 1)
f(18) = 4.90 + 4(17) = 72.90
So, the total cost of 18 battery packs is $72.90.
INTEGRATE TECHNOLOGY
Modeling Arithmetic Sequences From a Graph
Explain 2
Display the Example. Provide students with a domain
value and have a student circle that point on the
graph. Then ask the student to explain what the point
means in the context of the problem. Ask another
student to locate and plot the next value that would
appear on the graph.
Given a graph of a real-world situation involving an arithmetic sequence, you can construct a function model and use
it to solve problems.

Construct an explicit rule in function notation for the arithmetic
sequence represented in the graph, and use it to solve the problem.
D’Andre collects feather pens. The graph shows the number of feather pens
D’Andre has collected over time, in weeks. According to this pattern, how
many feather pens will D’Andre have collected in 12 weeks?
Represent the sequence in a table.
n
1
2
3
4
f(n)
18
37
56
75
Number of feather pens
Example 2
QUESTIONING STRATEGIES
80
70
60
50
40
30
20
10
0
Find the common difference.
y
(4, 75)
(3, 56)
(2, 37)
(1, 18)
x
1 2 3 4 5 6
Time (weeks)
d = 37 - 8 = 19
Use the general explicit rule for an arithmetic sequence to write the rule in function
notation. Substitute 18 for ƒ(1)and 19 for d.
ƒ(n) = ƒ(1) + d(n - 1)
© Houghton Mifflin Harcourt Publishing Company
How would a graph for the sequence
52, 48, 44, 40, … be similar to and different
from the graph in this exercise? The points of the
graph would still be in a straight line, but the points
would go from upper left to lower right to indicate a
sequence with a constant decrease rather than a
constant increase.
The table below shows the total cost, in dollars, of purchasing n battery packs.
ƒ(n) = 18 + 19(n - 1)
To determine the number of feather pens D’Andre will have collected after 12 weeks,
find ƒ(12).
ƒ(n) = 18 + 19(n - 1)
ƒ(12) = 18 + 19(11)
ƒ(12) = 18 + 209
ƒ(12) = 227
So, if this pattern continues, D’Andre will have collected 227 feather pens in 12 weeks.
Module 4
173
Lesson 3
COLLABORATIVE LEARNING
A1_MTXESE346734_U2M04L3 173
Small Group Activity
Have students work in groups of three. Have each student write a problem
involving an arithmetic sequence in a different real-world context, such as the cost
of one or more items of the same price. Each student solves the problem on a
separate sheet of paper. Students pass the problems to students on their right, then
solve the new problems using separate sheets of paper. When all students are
finished, they pass the problems to the right once more. When all students have
solved the problems, have them compare answers and discuss any differences.
173
Lesson 4.3
10/31/14 11:51 AM
Eric collects stamps. The graph shows the number of stamps that Eric has
collected over time, in months. According to this pattern, how many stamps
will Eric have collected in 10 months?
Number of stamps
B
Represent the sequence in a table.
n
1
2
20
f(n)
3
33
4
46
70
60
50
40
30
20
10
0
59
y
(4, 59)
(3, 46)
(2, 33)
(1, 20)
x
1 2 3 4 5
Time (months)
Find the common difference.
d = 33 - 20 = 13
Use the general explicit rule for an arithmetic sequence to write the rule in function notation.
Substitute 20 for ƒ(1) and 13 for d.
ƒ(n) = ƒ(1) + d(n - 1)
ƒ(n) = 20 +
13 (n - 1)
(
To determine the number of stamps Eric will have collected in 10 months, find ƒ
ƒ
ƒ(n) = ƒ(1) + d(n - 1)
(
10
)=
20 + 13
(
9
) = 137
So, if this pattern continues, Eric will have collected 137 stamps in
10
10
).
months.
Reflect
4.
How do you know which variable is the independent variable and which variable is the dependent variable
in a real-world situation involving an arithmetic sequence?
The independent variable indicates the term of the sequence so it will usually increase by
Your Turn
5.
The graph shows the height, in inches, of a stack of boxes on a table as the
number of boxes in the stack increases. Find the height of the stack with
7 boxes.
n
1
2
3
4
f(n)
34
51
68
85
d = 51 - 34 = 17
f(7) = 34 + 17(6) = 136
f(n) = 34 + 17(n-1)
Height (in.)
Construct an explicit rule in function notation for the arithmetic sequence
represented in the graph, and use it to solve the problem.
90
80
70
60
50
40
30
20
10
0
y
(4, 85)
(3, 68)
(2, 51)
(1, 34)
© Houghton Mifflin Harcourt Publishing Company
1 unit. The other variables have a constant difference between consecutive terms.
x
1 2 3 4 5
Number of boxes
So, the height of the stack with 7 boxes is 136 inches.
Module 4
174
Lesson 3
DIFFERENTIATE INSTRUCTION
A1_MTXESE346734_U2M04L3 174
Critical Thinking
10/31/14 11:51 AM
Have students think about the following question: Is it possible for two real-world
problems to generate the same graph? Explain. Invite students to give their
answers and explain their reasoning. Challenge them to come up with real-world
situations to support their answers. Yes, two problems generate the same graph
if both the domain and common difference are the same for both situations.
Modeling with Arithmetic Sequences
174
EXPLAIN 3
Quynh begins to save the same amount each month to save for a
future shopping trip. The graph shows total amount she has saved after
each month, n. What will be the total amount Quynh has saved after
12 months?
Modeling Arithmetic Sequences from
a Description
n
1
2
3
4
f(n)
250
300
350
400
f(n) = 250 + 50(n-1)
d = 300 - 250 = 50
f(12) = 250 + 50(11) = 800
QUESTIONING STRATEGIES
Amount saved ($)
6.
400
350
300
250
200
150
100
50
0
y
(4, 400)
(3, 350)
(2, 300)
(1, 250)
x
1 2 3 4 5 6
Number of months
So, Quynh will have saved a total of $800 after 12 months.
A student is getting better at math. On his first
test he scores 60 points, then he scores 65, 70,
74, and 80 on his next four tests. Can you calculate
how many points he will score on his eighth test?
Explain. No, because this is not an arithmetic
sequence. There is no common difference.
Explain 3
Modeling Arithmetic Sequences From a Description
Given a description of a real-world situation involving an arithmetic sequence, you can construct a function model
and use it to solve problems.
Construct an explicit rule in function notation for the arithmetic sequence represented,
and use it to solve the problem. Justify and evaluate your answer.
Example 3
The odometer on a car reads 34,240 on Day 1. Every day the car is driven 57 miles. If this pattern
continues, what will the odometer read on Day 15?
Analyze Information
• The odometer on the car reads 34,240 miles on Day 1.
• Every day the car is driven
57
miles.
ƒ(1) = 34,240 and
57
d=
© Houghton Mifflin Harcourt Publishing Company
Formulate a Plan
Write an explicit rule in function notation for the arithmetic sequence, and use it to
find f(15) , the odometer reading on Day 15.
Solve
ƒ(n) = ƒ(1) + d(n - 1)
ƒ
(
(
ƒ
15
15
ƒ(n) = 34,240 +
57
(n - 1)
34,240 +
57
14
)=
)=
35,038
(
)
On the Day 15, the odometer will show 35,038 miles.
Module 4
175
Lesson 3
LANGUAGE SUPPORT
A1_MTXESE346734_U2M04L3 175
Connect Context
As students work through this lesson, have them explain the new concepts in their
own words. Suggest that they work together to provide explanations that are as
complete as possible. Ask students to suggest connections to concepts they have
already learned and tell how those concepts are extended by the new material.
175
Lesson 4.3
10/31/14 11:51 AM
Justify and Evaluate
is
Using an arithmetic sequence model
ELABORATE
reasonable because the number of
miles on the odometer increases by the same amount each day.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
By rounding and estimation:
34,200 + 60(14) = 34,200 +
840
= 35,040 miles
So 35,038 miles is a reasonable answer.
Draw an arrow diagram representing a sequence. In a
set representing the domain, show the first few
position numbers of the sequence. In a set
representing the range, show the first few terms of the
sequence. Connect each position number with an
arrow to its corresponding term. Have students
identify the domain and the range of the function
representing the sequence.
Your Turn
Construct an explicit rule in function notation for the arithmetic sequence represented,
and use it to solve the problem. Justify and evaluate your answer.
7.
Ruby signed up for a frequent-flier program. She receives 3400 frequent-flier miles for the first round-trip
she takes and 1200 frequent-flier miles for all additional round-trips. How many frequent-flier miles will
Ruby have after 5 round-trips?
f(1) = 3400 and d = 1200
f(n) = 3400 + 1200(n - 1)
f(5) = 3400 + 1200(4) = 8200
8.
After 5 round-trips, Ruby will have 8200 frequent-flier miles.
A gym charges each member $100 for the first month, which includes a membership fee, and $50 per month
for each month after that. How much money will a person spend on their gym membership for 6 months?
f(1) = 100 and d = 50
f(6) = 100 + 50(5) = 350
f(n) = 100 + 50(n - 1)
SUMMARIZE THE LESSON
After 6 months, a person will have spent $350 on their gym membership.
What must be true about a real-world
problem in order to use arithmetic sequences
to solve it? The problem must describe a situation
that can be represented by a sequence that is
increasing or decreasing at a constant rate.
Elaborate
9.
What domain values usually make sense for an arithmetic sequence model that represents a real-world situation?
The domain is the set of term numbers or positions, so the domain values that make sense
are usually positive integers.
common difference when you know the difference between consecutive terms is constant,
find the difference between any two consecutive terms.
11. What are some ways to justify your answer when creating an arithmetic sequence model for a real-world
situation and using it to solve a problem?
You can make sure that the situation is accurately represented by the growth pattern of an
arithmetic sequence (repeated addition), and you can use estimation to make sure that the
answer is reasonable.
© Houghton Mifflin Harcourt Publishing Company
10. When given a graph of an arithmetic sequence that represents a real-world situation, how can you
determine the first term and the common difference in order to write a model for the sequence?
The first term is the y-coordinate of the point that has an x-coordinate of 1. To find the
12. Essential Question Check-In How can you construct a model for a real-world situation that involves an
arithmetic sequence?
Construct an explicit rule for the sequence in function notation, and specify reasonable
domain values.
Module 4
A1_MTXESE346734_U2M04L3 176
176
Lesson 3
17/02/14 5:01 PM
Modeling with Arithmetic Sequences
176
Evaluate: Homework and Practice
EVALUATE
1.
• Online Homework
• Hints and Help
• Extra Practice
A T-shirt at a department store costs $7.50. The total cost, in dollars,
of a T-shirts is given by the function C(a) = 7.5a.
a. Complete the table of values for 4 T-shirts.
T-shirts
1
Cost ($)
ASSIGNMENT GUIDE
$7.50
4
$30.00
Explore
Interpreting Models of Arithmetic
Sequences
Exercises 1–3
Example 1
Modeling Arithmetic Sequences
from a Table
Exercises 4–5,
15, 17
Example 2
Modeling Arithmetic Sequences
from a Graph
Exercises 6–9,
16, 19
Example 3
Modeling Arithmetic Sequences
from a Description
Exercises 10–14,
18
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Communication
Write or project the following problem. James is
given 75 vocabulary words the first week in English
class. He learns 10 words the first day and 5 more
each day after that. How many days will it take James
to learn all 75 vocabulary words? Invite volunteers to
take turns explaining how they would solve this
problem.
Each shirt costs $7.50, so the common difference is 7.5.
c. What does the variable a represent? What are the reasonable
domain values for a?
The variable a represents the number of T-shirts. Only nonnegative
integers make sense for a in this situation.
2.
A car dealership sells 5 cars per day. The total number
of cars C sold over time in days is given by the function
C(t) = 5t.
a. Complete the table of values for the first 4 days of
sales.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Tetra
Images/Corbis
Practice
Time (days)
1
2
3
4
Number of Cars
5
10
15
20
b. Determine the common difference.
Each day 5 cars are sold, so the common difference is 5.
c. What do the variables represent? What are the reasonable
domain and range values for this situation?
The variable C represents the total number of cars sold. The variable t represents
time, in days. The reasonable domain and range values are positive integers.
3.
A telemarketer makes 82 calls per day. The total number of calls made over time,
in days, is given by the function C(t) = 82t.
a. Complete the table of values for 4 days of calls.
Time (days)
1
2
3
4
Number of Calls
82
164
246
328
b. Determine the common difference.
Each day 82 calls are made, so the common difference is 82.
c. What do the variables represent? What are the reasonable domain and range
values for this situation?
The variable C represents the total number of calls over time, in days. The variable
t represents time in days. The reasonable values for both domain and range values is
positive integers.
Module 4
Exercise
A1_MTXESE346734_U2M04L3.indd 177
Lesson 4.3
3
$22.50
b. Determine the common difference.
Concepts and Skills
177
2
$15.00
177
Lesson 3
Depth of Knowledge (D.O.K.)
Mathematical Processes
2/22/14 2:44 PM
1–3
2 Skills/Concepts
1.A Everyday life
4–5
2 Skills/Concepts
1.A Everyday life
6–9
2 Skills/Concepts
1.A Everyday life
10–14
2 Skills/Concepts
1.A Everyday life
15
2 Skills/Concepts
1.A Everyday life
16
3 Strategic Thinking
1.A Everyday life
Construct an explicit rule in function notation for the arithmetic sequence represented in the table.
Then determine the value of the given term, and explain what it means.
4.
Darnell starts saving the same amount from each week’s paycheck. The table shows the total balance ƒ(n) of
his savings account over time in weeks.
Time (weeks) n
Savings Account Balance($) f (n)
1
2
3
4
$250
$380
$510
$640
Determine the value of ƒ(9), and explain what it represents in this situation.
f(n) = 250 + 130(n-1)
d = 130
f(9) = 250 + 130(8) = 1290
After 9 weeks, Darnell’s savings account will have a total of $1290.
Juan is traveling to visit universities. He notices mile markers along the road. He records the mile markers
every 10 minutes. His father is driving at a constant speed. Complete the table.
a.
Time Interval
Mile Marker
1
520
2
509
3
498
4
487
476
465
5
6
b. Find ƒ(10), and tell what it represents in this
situation.
d = -11
f(n) = 520 - 11(n - 1)
f(10) = 520 - 11(9)
f(10) = 421
The mile marker at time interval 10 is 421.
Construct an explicit rule in function notation for the arithmetic
sequence represented in the graph. Then determine the value of the
given term, and explain what it means.
6.
The graph shows total cost of a whitewater rafting trip and the
corresponding number of passengers on the trip. Find ƒ(8), and explain
what it represents.
1
75
2
100
3
125
4
150
f(n)
(4, 150)
(3, 125)
(2, 100)
(1, 75)
n
0
1 2 3 4 5 6
Number of passengers
f(n) = 75 + 25(n - 1)
d = 25
200
175
150
125
100
75
50
25
7.
n
f(n)
d = 15
1
20
2
35
3
50
4
65
5
80
f(n) = 20 + 15(n - 1)
Number of autographs
f(8) = 75 + 25(7) = 250
It will cost $250 for 8 passengers on the whitewater rafting trip.
Ed collects autographs. The graph shows the total number of autographs
that Ed has collected over time, in weeks. Find ƒ(12), and explain what
it represents.
f(12) = 20 + 15(11) = 185
0
At 12 weeks, Ed will have collected 185 autographs.
Module 4
Exercise
Depth of Knowledge (D.O.K.)
f(n)
(5, 80)
(4, 65)
(3, 50)
(2, 35)
(1, 20)
n
1 2 3 4 5 6
Time (weeks)
Lesson 3
178
A1_MTXESE346734_U2M04L3 178
90
80
70
60
50
40
30
20
10
© Houghton Mifflin Harcourt Publishing Company
Number of Passengers n
Total Cost ($) f(n)
Total cost ($)
5.
Mathematical Processes
17
2 Skills/Concepts
1.A Everyday life
18
2 Skills/Concepts
1.A Everyday life
19
3 Strategic Thinking
1.A Everyday life
2/22/14 3:56 AM
Modeling with Arithmetic Sequences
178
Finance Bob purchased a bus pass card with 320 points. Each week costs
20 points for unlimited bus rides. The graph shows the points remaining
on the card over time in weeks. Determine the value of ƒ(10), and explain
what it represents.
n
f(n)
1
300
d = -20
2
280
3
260
4
240
5
220
f(n) = 300 - 20(n - 1)
Points remaining
8.
300
f(n)
(3, 260)
225 (2, 280)
(4, 240)
(5, 220)
150
75
n
f(10) = 300 - 20(9) = 120
0
1 2 3 4 5
Time (weeks)
Biology The local wolf population is declining. The graph shows the local
wolf population over time, in weeks.
Find ƒ(9), and explain what it represents.
n
f(n)
1
100
2
94
3
88
4
82
5
76
Number of wolves
At 10 weeks, Bob will have 120 points remaining.
9.
f(n) = 100 - 6(n - 1)
d = -6
(1, 300)
f(9) = 100 - 6(8) = 52
f(n) (1, 100)
100
80 (2, 94)
60
(3, 88)
(4, 82)
40
(5, 76)
20
0
n
1 2 3 4 5
Time (weeks)
There will be 52 wolves in 9 weeks.
Construct an explicit rule in function notation for the arithmetic
sequence. Then determine the value of the given term, and explain
what it means.
10. Economics To package and ship an item, it costs $5.75 for the first pound and
$0.75 for each additional pound. Find the 12th term, and explain what it represents.
© Houghton Mifflin Harcourt Publishing Company
n
f(n)
1
5.75
d = 0.75
2
6.50
3
7.25
4
8.00
f(n) = 5.75 + 0.75(n - 1)
f(12) = 5.75 + 0.75(11) = 14
The total shipping cost of a 12-pound package is $14.00.
11. A new bag of cat food weighs 18 pounds. At the end of each day, 0.5 pound of food
is removed to feed the cats. Find the 30th term, and explain what it represents.
n
f(n)
1
18
d = -0.5
2
17.5
3
17
4
16.5
f(n) = 18 - 0.5(n - 1)
f(30) = 18 - 0.5(29) = 3.5
At the beginning of Day 30, the weight of the bag is 3.5 pounds.
Module 4
A1_MTXESE346734_U2M04L3 179
179
Lesson 4.3
179
Lesson 3
17/02/14 5:01 PM
12. Carrie borrows money interest-free to pay for a car repair. She will repay $120 monthly until
the loan is paid off. How many months will it take Carrie to pay off the loan? Explain.
n
f(n)
1
840
2
720
3
600
4
480
5
360
f(n) = 840 - 120(n - 1)
d = -120
Find the value of n for when f(n) = 0.
0 = 840 - 120(n - 1)
0 = 840 - 120n + 120
120n = 960
n=8
Therefore, it will take 8 months to pay off the loan.
13. The rates for a go-kart course are shown.
Number of Laps n
1
2
3
4
a. What is the total cost for 15 laps?
Total cost ($) f(n)
7
9
11
13
f(n) = 7 + 2(n - 1)
f(15) = 7 + 2(14)
f(15) = 35
The total cost of 15 laps is $35.00.
b. Suppose that after paying for 9 laps, the 10th lap is free. Will the sequence still be arithmetic? Explain.
No, it would not be an arithmetic sequence after the 9th term because the difference
between f(10) and f(9) would not be 2.
14. Multi-Part Seats in a concert hall are arranged in the pattern
shown.
a. The numbers of seats in the rows form an arithmetic
sequence. Write a rule for the arithmetic sequence.
Row 1 has 6 seats, Row 2 has 9 seats, Row 3 has 12 seats,
and so on.
f(n) = 6 + 3(n - 1)
Row 2
Row 3
© Houghton Mifflin Harcourt Publishing Company
d=3
Row 1
Row 4
b. How many seats are in Row 15?
f(15) = 6 + 3(14) = 48
There are 48 seats in Row 15.
c. Each ticket costs $40. If every seat in the first 10 rows is filled, what is the total
revenue from those seats?
f(1) + f(2) + f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9) + f(10)
= 6 + 9 + 12 + 15 + 18 + 21 + 24 + 27 + 30 + 33 = 195
195 · 40 = 7800
The total revenue will be $7800.
d. An extra chair is added to each row. Write the new rule for the arithmetic sequence
and find the new total revenue from the first 10 rows.
f(n) = 7 + 3(n - 1)
There will be 10 additional chairs: 195 + 10 = 205.
205 · 40 = 8200
The total revenue for the first 10 rows will be $8200.
Module 4
A1_MTXESE346734_U2M04L3 180
180
Lesson 3
2/22/14 3:58 AM
Modeling with Arithmetic Sequences
180
H.O.T. Focus on Higher Order Thinking
JOURNAL
15. Explain the Error The table shows the number of people who attend an
amusement park over time, in days.
Have students compare and contrast the way they
solve problems by modeling arithmetic sequences
from tables, graphs, and descriptions.
Time (days) n
1
2
3
4
Number of people f(n)
75
100
125
150
Sam writes an explicit rule for this arithmetic sequence: ƒ(n) = 25 + 75(n - 1)
He then claims that according to this pattern, 325 people will attend the amusement park on
Day 5. Explain the error that Sam made.
Sam did not write the explicit form of the sequence correctly. The formula should be
f(n) = 75 + 25(n - 1). Therefore, 175 people will attend the park on the Day 5 because
f(5) = 75 + 25(4) = 175.
16. Communicate Mathematical Ideas Explain why it may be harder to find the nth value of an
arithmetic sequence from a graph if the points are not labeled.
Without points labeled, it will be difficult to determine the exact values of the variables.
17. Make a prediction Verona is training for a marathon. The first part of her training schedule is given in
the table.
Session n
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Dudarev
Mikhail/Shutterstock
Distance (mi) f (n)
1
2
3
4
5
6
3.5
5
6.5
8
9.5
11
a. Is this training schedule an arithmetic sequence? Explain.
If it is, write an explicit rule for the sequence.
The training schedule is an arithmetic sequence
because the common difference between each pair of
consecutive sessions is 1.5 miles.
b. If Verona continues this pattern, during which training
session will she run 26 miles?
Find the session n for which f (n) is 26.
26 = 3.5 + 1.5(n - 1)
26 = 3.5 + 1.5n - 1.5
16 = n
Verona will run 26 miles on the sixteenth session.
18. If Verona’s training schedule starts on a Monday and she runs every third day, on which day
will she run 26 miles?
Monday, Thursday, Sunday, Wednesday, Saturday, Tuesday, Friday, Monday, Thursday,
Sunday, Wednesday, Saturday, Tuesday, Friday, Monday, Thursday
If she trains every third day and begins on a Monday, then she will run 26 miles on a
Thursday.
Module 4
A1_MTXESE346734_U2M04L3 181
181
Lesson 4.3
181
Lesson 3
2/22/14 4:00 AM
Time (months) n
Amount of money ($) f (n)
1
2
3
4
250
300
350
400
A person deposits $250 dollars into a bank account. Each month, he adds
$25 dollars to the account, and no other transactions occur in the account.
Amount of money ($)
19. Multiple Representations Determine whether the following graph,
table, and verbal description all represent the same arithmetic sequence.
400
350
300
250
200
150
100
50
0
No, the verbal description does not represent the same
sequence. The graph and table have a common difference of
$50, but the verbal description has a common difference of $25.
f(n)
(4, 400)
(3, 350)
(2, 300)
(1, 250)
QUESTIONING STRATEGIES
What do the bars in the graph represent? The
first four terms in the sequence, 215, 241,
267, and 293.
n
What does the difference in the heights of the
bars tell you? The difference in their heights
discloses the common difference of the sequence. In
this situation, it is 26.
1 2 3 4 5 6
Time (months)
Lesson Performance Task
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Communication
The graph shows the population of Ivor’s ant colony over the first four weeks. Assume the ant
population will continue to grow at the same rate.
a. Write an explicit rule in function notation.
b. If Ivor’s ants have a mass of 1.5 grams each, what will be the
total mass of all of his ants in 13 weeks?
a(n) = 215 + 26(n - 1)
a(13) = 215 + 26(13 - 1)
a(13) = 215 + 312
a(13) = 527
Number of Ants
Ivor’s Ant Farm
The common difference for each week is 26 ants.
Therefore, the explicit rule for this situation can be
written as a(n) = 215 + 26(n - 1).
300
Have students formulate an explanation of why the
explicit rule could also be a(n) = 189 + 26n.
Students should explain that a(n) = 189 + 26n is a
simplified version of a(n) = 215 + 26(n - 1).
293
275
267
250
241
225 215
200
0
1
2
3
4
Weeks
527 ∙ 1.5 g = 790.5 g
© Houghton Mifflin Harcourt Publishing Company
Therefore, in 13 weeks, the colony will have a mass of
about 790 grams.
c. When the colony reaches 1385 ants, Ivor’s ant farm will not
be big enough for all of them. In how many weeks will the ant
population be too large?
a(n) = 215 + 26(n - 1)
1385 = 215 + 26(n - 1)
1170 = 26(n - 1)
46 = n
Therefore, in 46 weeks the ant farm will not be big enough.
Module 4
182
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Lesson 3
EXTENSION ACTIVITY
A1_MTXESE346734_U2M04L3 182
Have students create a bar graph of an arithmetic sequence for the number of seats
in each of the first 4 rows of a theater. The first term should be the number of seats
in the longest row of seats. Students should trade bar graphs and determine how
many rows of seats there are in all, given a row must have at least 4 seats in it.
2/22/14 4:00 AM
Students should note that the explicit rule involves subtraction of (n – 1)d value
from the value of the first term since the number of seats is declining by a
constant difference.
Modeling with Arithmetic Sequences
182