Standards for Math Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning. © 2021 Saga Education
Sample Lesson & Activity
This lesson and activity are meant for reference
only and are not to be reproduced in any way.
Lesson: A6.8 Arithmetic Sequences
Activity: A6.8 Arithmetic Sequence Cards
Because our materials reach thousands of students across several districts, we
intentionally make them robust so they can be cut down and tailored to individual
students. We never expect a Saga Fellow to cover everything within a lesson. We
include an ample number of problems and tutoring strategies so Fellows can pick
and choose what works best to meet their students’ needs.
Activities are a way for students to dig into the concepts conceptually from a
different point of view. Our activities are intentionally more open-ended than our
problem sets, with many options for implementing to meet student needs.
Lesson A6.8 Arithmetic Sequences Page 1 of 18
© 2020 Saga Education
Objective A6.8: Identify and create arithmetic sequences.
F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model
situations, and translate between the two forms.
A6.8 Fellow Implementation Notes & Key
Key Points
(1) A sequence is an ordered list of numbers that creates a pattern.
(2) An arithmetic sequence is created by adding the same value, called the common difference, to each
term in order to determine the next term.
Lesson Overview
Students will learn that an arithmetic sequence is a list of numbers where a constant is being added to
each term in order to get the next term. The constant that is added to each term in the pattern is
called the common difference. Students will learn how to identify lists of numbers that are arithmetic
sequences. They will also learn how to model both visual patterns and real life situations using the
formula for arithmetic sequences, a
n
= a
1
+ d(n – 1).
Potential misconceptions and errors:
(1) Students may not understand the meaning of the components in the formula, a
n
= a
1
+ d(n – 1).
a
n
represents the n
th
term in the sequence, a
1
represents the first term in the sequence, and d
represents the common difference between subsequent terms. (n – 1) represents the term
number minus 1. Since a
1
(the first term) is already represented in the formula, we need to
subtract out the first term so that we do not count it twice. Have students write the formula in
their math notebook and label its components.
(2) Students may confuse the language they should use when discussing visual patterns vs.
discussing arithmetic sequences. Arithmetic sequences have terms and term numbers while
visual patterns have figures and figure numbers.
Key terms: arithmetic sequence, common difference, terms, patterns
Lesson A6.8 Arithmetic Sequences Page 2 of 18
© 2020 Saga Education
Background Skills/Do Now
(See Lesson F1.13 Patterns)
1. Find the next three terms in each pattern.
a) 2, 4, 6, 8, …
10, 12, 14
b) 12, 9, 6, 3, …
0, -3, -6
(See Lesson A6.7 Model & Solve Linear Functions)
2. Write an equation to model the following
data.
x
f(x)
1
5
2
9
3
13
f(x) = 1 + 4x
3. Review From Yesterday
4. Spiral Review
Lesson A6.8 Arithmetic Sequences Page 3 of 18
© 2020 Saga Education
Opening Critical Thinking Task
Purpose of the opening task: Explore and introduce arithmetic sequences.
1. Compare the two patterns. What is similar between the two? What is different?
Pattern one: 2, 4, 6, 8, …
Pattern two: 12, 9, 6, 3,…
They are both changing by a steady rate. The first one is increasing by 2 each time and the second is decreasing by 3.
2. Find the next three terms in pattern one and pattern two. (Hint: You may have already done this in
the Do Now!) Show or explain your reasoning.
Pattern one: 10, 12, 14
8 + 2 = 10, 10 + 2 = 12, 12 + 2 = 14
OR I added the two (the rate of change) to each prior term to get the next.
Pattern two: 0, -3, -6
3 – 3 = 0, 0 – 3 = -3, -3 – 3 = -6
OR I added -3 (or subtracted 3) to each prior term to get the next.
3. An arithmetic sequence is an ordered pattern of numbers where a constant is being added to each
term to determine the next term. Are the patterns arithmetic sequences? How do you know?
Both pattern one and pattern two are arithmetic, because they are changing by the same amount each time.
• Why is pattern two still an arithmetic sequence even though it’s decreasing? This can be seen as adding a
negative
4. The constant being added to each term in an arithmetic sequence is called the common difference.
What is the common difference of each arithmetic sequence? How did you find it? Why does it make
sense to call it the common difference?
The common difference for Pattern one is 2 and the common difference for Pattern two is -3. The common difference is
determined by finding the difference between any term and the previous: 4 – 2 = 6 – 4 = 8 – 6 = 2, and
9 – 12 = 6 – 9 = 3 – 6 = -3. It makes sense to call it the common difference because we find it by subtracting.
5. An arithmetic sequence can be represented by the formula: a
n
= a
1
+ d(n – 1), where d is the
common difference and a
1
is the first term. Write an equation for each pattern using this formula.
What do you think n and a
n
represent?
Pattern one: a
n
= 2 + 2(n – 1)
Pattern two: a
n
= 12 + -3(n – 1)
a
n
represents the value of the term, n.
• How do we determine the 10
th
term of the sequence for pattern one?
• What might be an advantage to using the formula to find terms for arithmetic sequences (instead of
writing it out using the pattern)?
Lesson A6.8 Arithmetic Sequences Page 4 of 18
© 2020 Saga Education
Example 1
Find the 110
th
term of the
sequence 5, 9, 13, ….
a
n
= 5 + 4(n – 1)
a
110
= 5 + 4(110 – 1)
a
110
= 5 + 4(109)
a
110
= 5 + 436
a
110
= 441
Purpose of Example: Determine an explicit equation for arithmetic sequence
and discuss characteristics of the explicit formula
Tutoring Strategies
Have students brainstorm possible strategies for solving. They may
suggest writing out 110 terms of the sequence. Although this is a valid
way of solving, it is inefficient. Students may also come up with other
adapted or modified equations for the sequence. For example, a
student may multiply 107 by 4 and then add 13. Push them to think
about the most efficient model that can be used to represent any
arithmetic sequence (the formula).
• How do we know this sequence is arithmetic? constantly increases by 4
• How could writing an equation to model the sequence help us?
Differentiation: Have students start by finding the 10
th
term in the
sequence manually. Then have them check that the equation they
wrote gives the same result.
Students were introduced to the formula a
n
= a
1
+ d(n – 1) in the Critical
Thinking Task. They can use this formula to find the 110
th
term.
• What is the common difference? What is the first term?
• Why is 1 being subtracted from n in the equation a
n
= a
1
+ d(n – 1)?
first term is already represented by a
1
Students may notice that they already modeled the sequence in this
problem with a linear function for the Do Now. Compare the arithmetic
sequence formula with the linear function for this sequence.
• How is the equation from this problem different or similar to the
equation from the Do Now?
o What happens when we simplify the arithmetic sequence
equation?
a
n
= 5 + 4n – 4 → a
n
= 4n + 1
o How is the formula different or similar to a linear equation?
• How are slope and the common difference related?
both rate of change
Extend the example: Have students discuss how they can find a missing
middle term in an arithmetic sequence instead of a future term.
• How would we find the missing term if we were given the arithmetic
sequence: 5, , 13?
13 + 5
2
= 9
Lesson A6.8 Arithmetic Sequences Page 5 of 18
© 2020 Saga Education
Example 2
a) Find the number of stars in the
10
th
figure of the pattern using an
equation to model the number of
stars.
Number of stars = 3 + 3(n – 1)
Number of stars = 3 + 3(10 – 1)
Number of stars = 3 + 3(9)
Number of stars = 3 + 27
Number of stars = 30 stars
b) Organize your data into the table
below and graph.
Figure
Number
Number
of Stars
1
3
2
6
3
9
10
30
Lesson A6.8 Arithmetic Sequences Page 6 of 18
© 2020 Saga Education
Practice Problems
Level 1
1. Fill in the blanks below using the word bank.
slope term linear common difference
x-axis pattern y-axis term number
An arithmetic sequence is an ordered pattern in
which a constant amount is added to each term
to obtain the next term. It can be modeled by a
graph, in which the common difference is
represented by the slope . On a graph, the
x-axis represents the term number , while the
y-axis represents the term .
Note to Fellows: The last sentence has two sections that
can be interchanged, as long as the x-axis is linked to term
number, and the y-axis is linked to the term.
2. Label each underlined part of the formula.
a
n
= a
1
+ d(n – 1)
3. Find the common difference for each
arithmetic sequence.
a) -34, -28, -22, -16, …
d = 6
b) 15, 10, 5, 0, …
d = -5
c) 0.9, 0.5, 0.1, -0.3, …
d = -0.4
d) 12, 22, 32, 42, 52, …
d = 10
e) 3, 8, 13, 18, …
d = 5
4. a) Complete the table for the sequence
modeled below.
Figure
Number
Number of
Dots
1
1
2
3
3
5
4
7
b) Find the common difference.
d = 2
Lesson A6.8 Arithmetic Sequences Page 7 of 18
© 2020 Saga Education
5. The following numbers are the start of an
arithmetic sequence with common difference
-20. Find the next four terms in the sequence.
Explain how you found your response.
25, 5, …
-15, -35, -55, -75
The common difference is -20, so if we continue to apply
this common difference by subtracting 20 four times, we
get -15, -35, -55, and -75.
6. a) The pattern below models a sequence. Find
the common difference between the number of
rings.
d = 1
b) Graph the sequence modeled by the pattern.
7. Find the common difference for the sequence
represented by each table.
a)
n
1
2
3
4
5
a
n
16
15
14
13
12
d = -1
b)
n
1
2
3
4
a
n
3
4
5
4
7
4
9
4
d =
2
4
!=
1
2
c)
n
1
2
3
4
5
a
n
2π
4π
6π
8π
10π
d = 2π
8. Determine the next three terms in the
sequence modeled in each figure:
a)
a
5
= 20; a
6
= 24; a
7
= 28
b)
a
5
= 12 smileys
a
6
= 14 smileys
a
7
= 16 smileys
Image sources: Big Ideas Math
Lesson A6.8 Arithmetic Sequences Page 8 of 18
© 2020 Saga Education
Level 2
9. Barrett says a common difference that’s a
negative means the sequence is decreasing. Is
he correct? If so, explain. If not, give a counter
example.
Yes, Barrett is correct. If the common difference is
negative, then we must add a negative number to get
from one term to the next term of the sequence, making
the numbers decrease.
10. Find the missing terms in each arithmetic
sequence.
a) 26 , 35, 44 , 53, …
common difference =
53 – 35
2
=
18
2
= 9
OR 35 + 2d = 53 → d = 9
b) 2, 6 , 10 , 14, ….
common difference =
14 – 2
3
=
12
3
= 4
OR 2 + 3d = 14 → d = 4
c) 3, 12 , 21, 30 , …
common difference =
21 – 3
2
=
18
2
= 9
OR 3 + 2d = 21 → d = 9
d) 3, 9 , 15 , 21, …
common difference =
21 – 3
3
=
18
3
= 6
OR 3 + 3d = 21 → d = 6
11. Find the 23
rd
term of the sequence:
19.5, 19.9, 20.3, 20.7, ….
19.9 – 19.5 = 20.3 – 19.5 = 20.7 – 20.3 = 0.4
Arithmetic sequence, common difference = 0.4
a
n
= 19.5 + 0.4(n – 1)
a
23
= 19.5 + 0.4(23 – 1)
a
23
= 28.3
12. A molecule of water has two hydrogen atoms
and one oxygen atom.
a) Write an equation to the model the total
number of atoms, a, based on the number of
molecules, n.
a
n
= 3 + 3(n – 1) OR a
n
= 3n
b) Find the number of atoms in 35 water
molecules.
a
35
= 3 + 3(35 – 1) = 3 + 3(34) = 105
Lesson A6.8 Arithmetic Sequences Page 9 of 18
© 2020 Saga Education
13. A side of an apartment building is shaped like
a staircase. The windows are arranged in
columns. How many windows are in the tallest
column if the apartment building has 15
columns?
Pattern: 2, 4, 6, 8, …
a
n
= 2 + 2(n – 1) OR a
n
= 2n
a
15
= 2 + 2(15 – 1) a
15
= 2(15) = 30
a
15
= 2 + 2(14)
a
15
= 2 + 28
a
15
= 30
14. Estephanie just passed her road test and got
her driver’s license. She works at a movie
theater part-time after school and wants to save
up to buy her own car. On the first week, she
saves $360 from her paycheck and then each
week after, she saves $200 of her paycheck for
her new car. How much money will Estephanie
have saved after 12 weeks?
a
n
= 360 + 200(n – 1)
a
12
= 360 + 200(12 – 1)
a
12
= 360 + 200(11)
a
12
= 360 + 2200
a
12
= 2560
Estephanie has $2,560 saved for her new car after 12
weeks.
Lesson A6.8 Arithmetic Sequences Page 10 of 18
© 2020 Saga Education
15. Trevor is volunteering with an organization
called Rock the Vote to help members of his
community register to vote. On the first day,
Trevor’s team helped 26 community members
register and pledge to vote in the upcoming
election. Every day after the first, they help 3
more people register and pledge to vote. How
many community members will be registered to
vote after 25 days?
a
1
= 26
d = 3
a
n
= 26 + 3(n – 1)
a
25
= 26 + 3(25 – 1)
a
25
= 26 + 3(24)
a
25
= 26 + 72
a
25
= 98
98 community members will be registered to vote after
25 days.
16. Nadia is at her community’s winter carnival.
She is excited because this year there is a Ferris
wheel, drop tower, and bumper cars. There is no
fee to enter the carnival. However, it costs
$3.00 for the first ride and then $1.50 for each
additional ride. Nadia has $20 to spend at the
carnival. If Nadia goes on 10 rides, will she have
enough money left over to buy funnel cake that
costs $2.50?
a
1
= 3
d = 1.5
a
n
= 3 + 1.5(n – 1)
a
10
= 3 + 1.5(10 – 1)
a
10
= 3 + 1.5(9)
a
10
= 3 + 13.5
a
10
= 16.5
Nadia will spend $16.50 after 10 rides.
$20 – $16.5 = $3.50 so she will have enough money to
buy the funnel cake.
17. Matthias solved the following problem as
shown. Did he do it correctly? If so, explain his
work. If not, explain and correct his error.
Fill in the missing term in the arithmetic sequence
2.2, , 6, ….
2.2 + 6
2
! = !
2.8
2
! = !1.4
The missing term is 1.4.
Matthias set up the problem correctly using the midpoint
formula, because the missing term must be exactly in the
middle of 2.2 and 6. However, he incorrectly added 2.2
and 6. His answer should have been:
2.2 + 6
2
!=!
8.2
2
!=!4.1
!
We can check our answer by finding the common
difference between our three terms:
4.1 – 2.2 = 1.9
6 – 4.1 = 1.9
ü
18. Diana says you only need two sequential
terms of any arithmetic sequence to be able to
write an equation for the sequence. Do you
agree or disagree? Explain or show your
reasoning.
Diana is right, given that we know that it is an arithmetic
sequence. With two sequential terms, we can find the
common difference. For example, for the sequence 4, 8, …
d = 4 and a
1
= 4. However, if we are not told that this is
an arithmetic sequence, then the pattern may be to
multiply by 2, making the sequence 4, 8, 16, 32, …
Without knowing the type of sequence, we could not write
an equation to accurately model it.
Lesson A6.8 Arithmetic Sequences Page 11 of 18
© 2020 Saga Education
19. Determine if each models an arithmetic
sequence and explain your reasoning. If it is,
write the equation of the sequence the pattern is
modeling.
a)
This models an arithmetic
sequence. It decreases
by 15 between each
term.
a
n
= 70 + -15(n – 1)
b)
This does not model an
arithmetic sequence,
because it does not have
a single common
difference.
c)
This models an arithmetic
sequence. It increases by
4 between each term.
a
n
= 4 + 4(n – 1)
OR
a
n
= 4n
Image sources: Big Ideas Math
20. Determine if each pattern models an
arithmetic sequence and explain your reasoning.
If it is, write the equation of the sequence the
pattern is modeling.
a)
This does not model an arithmetic sequence. Even though
we can see that it grows in a predictable manner, there is
no common difference. It grows 5 squares, then 7.
b)
This models an arithmetic sequence. It grows by 4 both
times. a
n
= 1 + 4(n – 1)
Lesson A6.8 Arithmetic Sequences Page 12 of 18
© 2020 Saga Education
Level 3
21. Jeremiah is using the language learning app
Duolingo to try and learn Spanish for fun.
During the first week he learns 40 new phrases.
After the first week, he learns 25 new phrases
per week. How many weeks will it take
Jeremiah to learn 3,015 phrases in Spanish?
a
1
= 40
common difference = 25
a
n
= 40 + 25(n – 1)
3015 = 40 + 25n – 25
3015 = 15 + 25n
3000 = 25n
120 = n
It will take Jeremiah 120 weeks to learn 3,015 phrases.
22. Your school’s auditorium has 20 rows total.
The first row has 22 seats. The number of seats
in each row increases by 6 as you move towards
the back of the auditorium. For your graduation,
your entire family wants to sit together in the
last row.
a) How many family members are you inviting to
your graduation?
Answers will vary.
b) If 131 seats in the last row of the auditorium
are already taken by other families, will your
entire family be able to sit together in the last
row? Justify your answer.
a
1
= 22
common difference = 6
a
n
= 22 + 6(n – 1)
a
20
= 22 + 6(20 – 1)
a
20
= 22 + 6(19)
a
20
= 136
136 – 131 = 5 seats are available in the last row
Note to Fellows: If students invite 5 or less family
members, then their entire family will be able to sit in the
last row together.
Lesson A6.8 Arithmetic Sequences Page 13 of 18
© 2020 Saga Education
23. Gino is planning a big birthday party for his
friend Angelina. He needs to rent enough tables
for all of the guests. 1 square table seats 4
people. When 2 square tables are pushed
together it seats 6 people.
a) How many people would 3 square tables that
are pushed together seat? Show or explain your
reasoning.
There are 3 people at each end table and 2 people seated
in the middle. 3 tables pushed together seats 8 people.
b) How many tables would Gino need to push
together in order to seat 105 guests at the
birthday party?
Pattern: 4, 6, 8...
a
n
= 4 + 2(n – 1)
105 = 4 + 2n – 2
105 = 2 + 2n
103 = 2n
n = 51.5
Gino needs to rent 52 tables to seat 105 guests
c) Will there be any extra seats? If so, how
many?
a
n
= 4 + 2(n – 1)
a
52
= 4 + 2(52 – 1)
a
52
= 4 + 2(51)
a
52
= 4 + 102
a
52
= 106
There will be 1 extra seat.
24. Yolanda is trying to become an influencer on
Instagram. After the first month, Yolanda will
have 30,000 followers. Every month after the
first, she gains 1,250 more followers.
a) How many followers will Yolanda have after 6
months?
a
1
= 30,000
d = 1,250
a
n
= 30,000 + 1,250(n – 1)
a
6
= 30,000 + 1,250(6 – 1)
a
6
= 30,000 + 1,250(5)
a
6
= 30,000 + 6,250
a
6
= 36,250
In 6 months, Yolanda will have 36,250 followers on
Instagram.
b) Yolanda wants to be an influencer for Nike.
She needs to have 100,000 Instagram followers
before Nike will partner with her. How many
months will it take Yolanda to have enough
followers to become an influencer for Nike?
a
n
= 30,000 + 1,250(n – 1)
100,000 = 30,000 + 1,250(n – 1)
100,000 = 30,000 + 1,250n – 1,250
100,000 = 28,750 + 1,250n
71,250 = 1,250n
57 = n
It will take Yolanda 57 months to have enough followers
to become an influencer for Nike.
Lesson A6.8 Arithmetic Sequences Page 14 of 18
© 2020 Saga Education
25. Create your own arithmetic sequence with a
common difference of 4.5. List at least 3 terms.
Explain how we know your sequence is an
arithmetic sequence.
Answers will vary. Sample response:
1, 5.5, 10, …
26. Describe the 15
th
figure for each pattern.
a)
Number of Sides = 3 + 1(n – 1); a
15
= 3 + 1(15 – 1) = 17
The 15
th
figure will be a 17
th
sided regular polygon.
b)
Diameters Drawn = 1 + 1(n – 1);
a
15
= 1 + 1(15 – 1) = 15
The 15
th
figure will be a circle with 15 diameters drawn in.
Lesson A6.8 Arithmetic Sequences Page 15 of 18
© 2020 Saga Education
27. This month Juanita listened to 12 new songs
on Spotify because she wants to expand her
music collection. She decides that she will listen
to 15 new songs every month. How long will it
take Juanita to listen to at least 500 new songs?
a
1
= 12
common difference = 15
a
n
= 12 + 15(n – 1)
500 = 12 + 15n – 15
500 = -3 + 15n
503 = 15n
33.5 = n
It will take Juanita 34 months to listen to at least 500
new songs.
28. Approximately 1.6 million people in the
United States do not have regular access to
clean drinking water and sanitation necessities
such as toilets and showers. However, it takes
1,800 gallons of water to produce 1 pound of
beef.
a) Every McDonald’s Big Mac uses one-half
pound of beef. If one McDonald’s restaurant in
Washington DC sells 148 Big Macs today and
then 175 Big Macs every day after, then how
many gallons of water will the McDonald’s use
after 1 week?
1,800 gallons per pound of beef
1,800(0.5) = 900 gallons per half pound of beef
Day 1: 148(900) = 133,200 gallons
Common difference: 175(900) = 157,500 gallons
a
n
= 133,200 + 157,500(n – 1)
a
7
= 133,200 + 157,500(7 – 1)
a
7
= 1,078,200
The McDonald’s will use 1,078,200 gallons of water in 1
week.
b) Across the entire United States, an average of
1.5 million Big Macs are sold every day. How
many gallons of water does the U.S. use for Big
Macs every day?
1,500,000(0.5) = 750,000 pounds of beef per day
1,800(1,750,000) = 1,350,000,000
1.35 billion gallons of water is used for Big Macs every
day in the United States.
c) Are you surprised by your answer to part b)?
Do you view the water footprint left by the meat
industry as a problem? Explain why or why not.
Answers will vary.
Lesson A6.8 Arithmetic Sequences Page 16 of 18
© 2020 Saga Education
29. a) Kiara says that the range of an arithmetic
sequence is always going to contain only positive
and negative integers. She reasons that since
the domain is composed of positive integers and
the sequence is increasing or decreasing at a
constant rate then the outputs will also be
integer values. Is Kiara correct? Explain.
She is not correct. While the common difference is a
constant rate, it is not necessarily an integer value. For
example, for the sequence a
n
= 19.5 + 0.4(n – 1), the
domain is positive integers, but the range has decimal
values.
b) Under what conditions would the range be
composed only of integers?
The outputs would be only integers if both the first term
and the common difference are also integers.
30. Over the last ten years the number of inches
of snow a town received formed an arithmetic
sequence. 21 inches of snow fell 10 years ago
and 19 inches fell 9 years ago.
a) What is the common difference?
Common difference = 19 – 21 = -2
b) Write a formula to model this sequence.
Answers will vary. Sample response:
If we let 10 years ago be represented by n = -10:
a
n
= 21 – 2(n + 10)
Note to Fellow: Students may define n differently and
create other valid formulas. This will affect the work
shown in parts c) and d), but will not affect the final
answers.
c) How many inches fell two years ago?
Two years ago is represented by n = -2:
a
-2
= 21 – 2(-2 + 10) = 21 – 2(8) = 21 – 16 = 5
Two years ago, five inches of snow fell.
d) When will snow stop falling according to this
model? What does this tell you about the
validity of this model?
0 = 21 – 2(n + 10)
0 = 21 – 2n – 20
0 = 1 – 2n
n =
1
2
According to the model, snow will stop falling in half a
year in the town. This does not seem very likely though,
so the model may not be accurate for this situation.
Lesson A6.8 Arithmetic Sequences Page 17 of 18
© 2020 Saga Education
31. Korina donates the same amount of money
each year to help protect the rainforest. At the
end of the second year, she has donated enough
money to protect 8 acres. At the end of the
third year, she has donated enough to protect 12
acres.
a) Does this represent an arithmetic sequence?
How do you know?
Yes. We can use the context to conclude that since Korina
donates the same amount each year, her money protects
the same number of additional acres each year, and the
sequence is arithmetic.
b) If Korina created a graph showing years vs
number of acres saved, what would the graph
look like? Explain how you know.
The graph would be linear, with a slope of 4 and a y-
intercept of 0. This makes sense because the number of
years and the number of acres saved are proportional.
Before she started donating, she saved 0 acres, and each
year she donates, she saves 4 more acres.
c) How many acres will her donations protect by
the end of the tenth year?
Common difference = 12 – 8 = 4
a
n
= 0 + 4n
OR a
n
= 4 + 4(n – 1)
OR a
n
= 8 + 4(n – 2)
OR a
n
= 12 + 4(n – 3)
a
10
= 4(10) = 40 acres
At the end of her tenth year, she will have saved 40 acres
of rainforest.
32. Use the sequences below to answer the
questions.
i. 2, 4, 6, 8…
ii. -5, -10, -15, -20…
iii. 7.5, 15, 22.5…
a) Write an equation to model each.
i. a
n
= 2 + 2(n – 1) = 2n
ii. a
n
= -5 – 5(n – 1) = -5n
iii. a
n
= 7.5 + 7.5(n – 1) = 7.5n
b) Zhang correctly says we do not need to know
about sequences to model this problem with an
equation, although we could. What does he
mean by this?
We have learned to model and write linear equations, so
we could just write a linear equation to model each set of
numbers without the use of the sequence formula.
c) What do you notice about the relationship
between the first term and the common
difference?
They are the same.
d) What would these sequences have in common
if they were graphed?
They all go through the origin and are linear.
e) These arithmetic sequences can be modeled
by proportional functions. How would you
describe proportional functions?
Proportional functions have a constant rate of change,
such that f(x) will change directly in proportion to the
change in x. The equation can be modeled by f(x) = kx,
where k is the rate of change. When graphed, they will
cross through the origin and are linear.
Lesson A6.8 Arithmetic Sequences Page 18 of 18
© 2020 Saga Education
Challenge
33. A geometric sequence is an ordered pattern
of numbers where a constant is being multiplied
by each term to determine the next term. The
constant being multiplied by each term in a
geometric sequence is called the common ratio.
For example, for the sequence: 2, 4, 8, 16, …
The common ratio is 2.
Find the common ratio for each:
a) 3, 9, 27, 81…
common ratio: 3
b) 8, -16, 32…
common ratio: -2
c) 100, 50, 25…
common ratio:
1
2
or 0.5
34. Find the missing terms in each geometric
sequence. Determine the common ratio.
a) 4, 8 , 16, 32 , …
4r
2
= 16 → common ratio = 2
b) 3, 6 , 12, 24 , …
3r
2
= 12 → common ratio = 2
c) 1, 2 , 4 , 8, …
1r
3
= 8 → common ratio = 2
Ticket To Leave
1. a) Determine if the sequence is arithmetic.
Explain.
5, 10, 15, 20, …
It is arithmetic because a constant (5) is begin added to
each term.
b) Determine the 41
st
term using an equation.
a
41
= 5 + 5(41 – 1) = 205
2. Determine the next three terms of the
sequence modeled in the table below.
n
1
2
3
4
5
a
n
25
23.5
22
20.5
19
a
6
= 17.5
a
7
= 16
a
8
= 14.5
b) What is the common difference? How do you
know?
d = -1.5
a
2
– a
1
= a
3
– a
2
= a
4
– a
3
= a
5
– a
4
= -1.5
3. Fellow’s Choice
4. Fellow’s Choice
Activity – Arithmetic Sequence Cards Page 1 of 5
© 2020 Saga Education
Arithmetic Sequence Cards Activity
This activity aligns with lesson A6.8 Arithmetic Sequences.
Cut out the card sets. Use as many or as few of the pieces (sequence, table, graph, description,
equation, situation) as you would like, depending on the needs of your students.
Option 1: Mix all cards together and match the sets. This does not need to include all 6 cards in each
set. You could choose to have students only match sequences to graphs. Or sequences to
descriptions and situations. If you choose to only use the table, graph, equation, and situation, these
cards could be applied to linear equations.
• Which was the hardest piece to match? Which were the easiest to pair up?
• Which situations (scenarios) could continue linearly forever in real life? Which wouldn’t? When might
they stop being a linear/arithmetic relationship?
Option 2: Draw one card from the set and come up with the other pieces. To do this as a team, have
partners each pick a different card from a set and come up with the missing pieces. This can be done
using teamwork, or done individually and checked as a team after. For example, Eveline picks a table
card so she has to find the sequence, graph, description, equation, and situation that go with her table.
From the same set of 6 cards, Malone chooses the graph card, so he has to find the sequence, table,
description, equation, and situation that match his graph. After completing these, Eveline and Malone
compare their answers.
• Which piece should always be identical? sequence, table, graph, equation
•
Which pieces could vary student to student? description could vary slightly, situation could vary drastically
Option 3: Look for patterns in the graphs using only the sequence and graph cards.
• How does the first term in the sequence affect the graph?
o How do you know if the graph starts above or below the x-axis?
• How does the common difference affect the graph?
o How do you know if the graph slopes up or down?
Option 4: Relate the sequence to the equation by using only the sequence and equation cards.
• Based on the sequence, how do we know if the slope, m, in the equation y = mx + b should be positive
or negative?
• Using the equation y = mx + b, is the constant b ever part of the sequence? Explain.
o How do we determine b based on the sequence?
• In the equation y = mx + b, how does the sequence affect the value of m? How does it affect the value
of b?
Option 5: Relate all cards by highlighting similar components. This can be done by first sorting the
cards into groups of 6, or by using the sets of 6 without cutting the cards apart. Use two different
colors to highlight/circle where we see the common difference and first term in each representation.
If your students have already covered slope-intercept form of equations, you can also determine
where you see the slope and the y-intercept in each representation, using a third color for y-intercept.
(Students should determine slope is the same as the common difference, so it does not need a fourth
color.)
• Where do we see the first term in each representation?
o Where is it easiest to locate? hardest? Do you ever have to do a calculation to find it?
• Where do we see the common difference in each representation?
o Where is it easiest to locate? hardest? Do you ever have to do a calculation to find it?
Activity – Arithmetic Sequence Cards Page 2 of 5
© 2020 Saga Education
Sequence
7, 10, 13, 16, 19, …
Table
x
y
1
7
2
10
3
13
4
16
5
19
Graph
Description
Terms increase by 3,
starting at 7
Equation
y = 3x + 4
Situation
Miriam earns $3 a day
to feed her neighbor’s
cat. After the first day
she had $7 in her
wallet.
Sequence
4, 7, 10, 13, 16, …
Table
x
y
1
4
2
7
3
10
4
13
5
16
Graph
Description
Terms increase by 3,
starting at 4
Equation
y = 3x + 1
Situation
Balzo earns $3 per week to
mow the grass or rake
leaves as a chore. After he
collected his money on the
first day, he had $4.
Activity – Arithmetic Sequence Cards Page 3 of 5
© 2020 Saga Education
Sequence
-7, -4, -1, 2, 5, …
Table
x
y
1
-7
2
-4
3
-1
4
2
5
5
Graph
Description
Terms increase by 3,
starting at -7
Equation
y = 3x – 10
Situation
Jordin earns $3 each day
she walks a neighbor child
to school. Before she
started walking children to
school, she was $10 in
debt to her brother.
Sequence
3, 5, 7, 9, 11, …
Table
x
y
1
3
2
5
3
7
4
9
5
11
Graph
Description
Terms increase by 2,
starting at 3
Equation
y = 2x + 1
Situation
Lupe likes to travel. Each
year she travels to two
more states. She started
only having visited her
state.
Activity – Arithmetic Sequence Cards Page 4 of 5
© 2020 Saga Education
Sequence
3
2
, 2,
5
2
, 3,
7
2
, …
Table
x
y
1
1.5
2
2
3
2.5
4
3
5
3.5
Graph
Description
Terms increase by
1
2
,,
starting at 1
Equation
y =
1
2
x + 1
Situation
Oscar likes botany so
planted a one-inch-tall
flower. Each week the
flower grew half an inch,
so it was 1.5 inches tall
after 1 week.
Sequence
-1, -3, -5, -7, -9, …
Table
x
y
1
-1
2
-3
3
-5
4
-7
5
-9
Graph
Description
Terms decrease by 2,
starting at -1
Equation
y = –2x + 1
Situation
Zenobia’s bank account
charges her $2 a day for
every day her account
balance is under $10. On
the first day she noticed
she was charged, she was
in debt $1.
Activity – Arithmetic Sequence Cards Page 5 of 5
© 2020 Saga Education
Sequence
1
2
, 0, −
1
2
, –1, −
3
2
, …
Table
x
y
1
0.5
2
0
3
-0.5
4
-1
5
-1.5
Graph
Description
Terms decrease by
1
2
,
starting at
1
2
Equation
y = −
1
2
x + 1
Situation
Bashir is playing a game
where for every bad move
he loses half a point. After
his first bad move he has
only half a point left, and
he continues to make only
bad moves.
Extension:
This GeoGebra widget allows students to explore the relationship between the first term, common
difference, and the graph of a sequence: https://www.geogebra.org/m/kpAAYWc4