LESSON
12.1
Name
Understanding
Geometric Sequences
Class
Date
12.1 Understanding Geometric
Sequences
Essential Question: How are the terms of a geometric sequence related?
Texas Math Standards
A1.12.D…write a formula for the nth term of…geometric sequences, given the value of
several of their terms.
The student is expected to:
Explore 1
A1.12.D
Write a formula for the nth term of arithmetic and geometric sequences,
given the value of several of their terms.
Exploring Growth Patterns of Geometric
Sequences
The sequence 3, 6, 12, 24, 48, … is a geometric sequence. In a geometric sequence, the ratio of successive terms is
constant. The constant ratio is called the common ratio, often represented by r.
Mathematical Processes

A1.1.F
Complete each division.
6= 2
_
3
12 = 2
_
6
48 = 2
_
24
24 = 2
_
12
Analyze mathematical relationships to connect and communicate
mathematical ideas.

The common ratio r for the sequence is 2 .
Language Objective

Use the common ratio you found to identify the next term in the geometric sequence.
3.F, 3.H, 4.G
The next term is 48 ·
Explain to a partner how to tell whether a sequence is a geometric
sequence.
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo
and examples of payment plans students might use
when charging for odd jobs. Then preview the Lesson
Performance Task.
© Houghton Mifflin Harcourt Publishing Company
The terms of a geometric sequence are related by a
common ratio, often represented by r.
2 = 96 .
Reflect
1.
Suppose you know the twelfth term in a geometric sequence. What do you need to know to find the
thirteenth term? How would you use that information to find the thirteenth term?
You need to know the common ratio, r. You can multiply the twelfth term by r.
2.
Discussion Suppose you know only that 8 and 128 are terms of a geometric sequence. Can you find the
term that follows 128? If so, what is it?
Only if you know that 8 and128 are successive terms. In that case, the common ratio is 16, and
ENGAGE
Essential Question: How area the
terms of a geometric sequence related?
Resource
Locker
the next term is 2048. However, 8 and 128 could be terms of a different geometric sequence.
For example, in the geometric sequence 8, 16, 32, 64, 128, ..., the next term is 256.
Explore 2
Comparing Growth Patterns of Arithmetic
and Geometric Sequences
Recall that in arithmetic sequences, successive (or consecutive) terms differ by the same nonzero number d, called
the common difference. In geometric sequences, the ratio r of successive terms is constant. In this Explore, you will
examine how the growth patterns in arithmetic and geometric sequences compare. In particular, you will look at the
arithmetic sequence 3, 5, 7, ... and the geometric sequence 3, 6, 12, ... .
Module 12
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Lesson 1
545
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Lesson 1
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on
patterns in geometric sequence
the comm
how the growth
and the
examine
3, 5, 7, ...
545
sequence
arithmetic
Module 12
545
Lesson 12.1
L1 545
9_U5M12
SE35387
A1_MTXE
2/14/15
12:11 PM
2/14/15 12:11 PM
The tables shows the two sequences.
3, 5, 7, ...
Term Number
1
2
3
4
5
EXPLORE 1
3, 6, 12, ...
Term
3
5
7
9
11
Term Number
1
2
3
4
5
Term
3
6
12
24
48
Exploring Growth Patterns of
Geometric Sequences
A
The common difference d of the arithmetic sequence is 5 - 3 = 2. The common ratio r of the geometric
6 =
sequence is _
2 .
3
B
Complete the table. Find the differences of successive terms.
INTEGRATE TECHNOLOGY
Have students complete the Explore activity in either
the book or online lesson.
Arithmetic: 3, 5, 7, ...
Term
Difference
CONNECT VOCABULARY
Term Number
1
3
—
2
5
5-3=
3
7
7-5= 2
4
9
9-7=
5
11
11 - 9 = 2
Make sure that students understand the meanings of
successive terms and ratio of successive terms. You can
explain that two successive terms are two terms that
are next to each other in the sequence. Have students
give examples of pairs of successive terms. Explain
that successive terms can also be called
consecutive terms.
2
2
Geometric: 3, 6, 12, ...
Term
Difference
Term Number
3
—
2
6
6-3=
3
12
12 - 6 = 6
4
24
24 - 12 = 12
5
48
48 - 24 = 24
Module 12
3
546
EXPLORE 2
© Houghton Mifflin Harcourt Publishing Company
1
Comparing Growth Patterns of
Arithmetic and Geometric Sequences
QUESTIONING STRATEGIES
How are the graphs of geometric sequences
and arithmetic sequences alike? How are they
different? Possible answer: They can both be
represented by a function with a domain that is the
set of positive integers, or a subset of consecutive
positive integers beginning with 1. The graph of a
geometric sequence follows a curve, while the
graph of an arithmetic sequence is linear.
Lesson 1
PROFESSIONAL DEVELOPMENT
A1_MTXESE353879_U5M12L1 546
Learning Progressions
22/02/14 11:57 AM
In an earlier module, students studied arithmetic sequences and wrote general
recursive and explicit rules for them. Students used these rules to solve real-world
problems involving arithmetic sequences. In this module, students will learn
about geometric sequences and exponential functions. In the next module,
students will learn more about exponential functions, including exponential
growth and decay functions.
Understanding Geometric Sequences
546
C
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
Compare the growth patterns of the sequences based on the tables.
For the arithmetic sequence, the differences are equal. The terms of the arithmetic
sequence increase by a fixed amount. For the geometric sequence, the differences
increase. The terms of the geometric sequence increase by an increasing amount.
Show students a graph of the first 4 terms of
f(n) = 2 + 2(n - 1) and f(n) = 2 · 2n - 1. Point out
that the shape formed by the points indicates whether
the graph represents an arithmetic or geometric
sequence. Look at three or more points before
determining whether the graph is linear or
exponential, since arithmetic and geometric
sequences can have the same first two terms.
D
Graph both sequences in the same coordinate plane. Compare the growth patterns based on
the graphs.
Sequence Patterns
50
y
Term
40
30
Geometric
sequence
20
10
EXPLAIN 1
Arithmetic
sequence
x
0
Extending Geometric Sequences
1
4
2
3
Term number
5
For the arithmetic sequence, the slope is constant, so the vertical distances between
successive points are the same. For the geometric sequence, the vertical distances
AVOID COMMON ERRORS
Reflect
3.
© Houghton Mifflin Harcourt Publishing Company
When finding a common ratio, students might divide
in the wrong order. Tell them they are really finding
what each term is being multiplied by to get the next
term, so the inverse operation (division), will
produce r. For example, when finding the common
1
, ...,
ratio of the geometric sequence 16, 4, 1, _
4
divide 4 by 16.
between the points increase by increasing amounts.
Which grows more quickly, the arithmetic sequence or the geometric sequence?
The geometric sequence
Explain 1
Extending Geometric Sequences
In Explore 1, you saw that each term of a geometric sequence is the product of the preceding term and the common
ratio. Given terms of a geometric sequence, you can use this relationship to write additional terms of the sequence.
Finding a Term of a Geometric Sequence
For n ≥ 2, the nth term, ƒ(n), of a geometric sequence with common ratio r is
ƒ(n) = ƒ(n - 1)r.
Module 12
547
Lesson 1
COLLABORATIVE LEARNING
A1_MTXESE353879_U5M12L1.indd 547
Peer-to-Peer Activity
11/18/14 11:41 PM
Have students work in pairs. Have one student write and graph a geometric
sequence for which the common ratio r is greater than 1. Have the other student
write and graph a geometric sequence for which the common ratio r is 0 < r < 1.
Have the students compare the graphs. Have the students switch roles and repeat
the exercise.
547
Lesson 12.1
Example 1

Find the common ratio r for each geometric sequence and use r to find the
next three terms.
QUESTIONING STRATEGIES
In giving the formula for the nth term of a
geometric sequence, why does the book say
“For n ≥ 2”? It is understood that n is an integer.
The first term of the sequence is f (1). For n = 0 and
1, f(n - 1) is not defined.
6, 12, 24, 48, …
12 = 2, so the common ratio r is 2.
_
6
For this sequence, ƒ(1) = 6, ƒ(2) = 12, ƒ(3) = 24, and ƒ(4) = 48.
ƒ(4) = 48, so ƒ(5) = 48(2) = 96.
ƒ(5) = 96, so ƒ(6) = 96(2) = 192.
ƒ(6) = 192, so ƒ(7) = 192(2) = 384.
The next three terms of the sequence are 96, 192, and 384.

100, 50, 25, 12.5, …
50 = 0.5 , so the common ratio r is
_
100
0.5 .
For this sequence, ƒ(1) = 100, ƒ(2) = 50, ƒ(3) = 25, and ƒ(4) = 12.5.
ƒ(4) = 12.5, so ƒ(5) = 12.5 (0.5) = 6.25 .
ƒ(5) = 6.25 , so ƒ(6) = 6.25 (0.5) =
ƒ(6) =
3.125 , so ƒ(7) =
3.125 .
3.125 (0.5) = 1.5625 .
The next three terms of the sequence are 6.25 ,
3.125 , and 1.5625 .
Reflect
Communicate Mathematical Ideas A geometric sequence has a common ratio of 3. The 4th term
is 54. What is the 5th term? What is the 3rd term?
The 5th term is 3 times the 4th, or 162. The 4th term, 54, is 3 times the 3rd term, so the 3rd
term is 18.
Your Turn
Find the common ratio r for each geometric sequence and use r to find the next three terms.
5.
5, 20, 80, 320, ...
6.
20
_
=4=r
5
f(4) = 320, so f(5) = 320(4) = 1280.
f(5) = 1280, so f(6) = 1280(4) = 5120.
1 , ...
9, -3, 1, -_
3
-3
1
=- =r
3
9
1
f(4) = - 1 , so f(5) = - 1 - 1 = .
9
3
3
3
_
f(5) =
f(6) = 5120, so f(7) = 5120(4) = 20,480.
The next three terms are 1280, 5120,
and 20,480.
_
_
_( _) _
1 .
_1 , so f(6) = _1 (-_1 ) = -_
9
_
9
3
27
1
_( _ ) = _
.
f(6) = - 1 , so f(7) = - 1 - 1
27
27
3
81
1
1
1 , and _
The next three terms are _, -_
.
9
Module 12
© Houghton Mifflin Harcourt Publishing Company
4.
548
27
81
Lesson 1
DIFFERENTIATE INSTRUCTION
A1_MTXESE353879_U5M12L1 548
Multiple Representations
13/11/14 4:45 PM
Help students making connections between a table representing an arithmetic
sequence and a graph of the sequence. For example, start with the table and have
students make the graph. Then start with the graph and have students make the
table. Do the same for a geometric sequence. Have students compare the graphs of
the two sequences.
Understanding Geometric Sequences
548
Recognizing Growth Patterns of Geometric
Sequences in Context
Explain 2
EXPLAIN 2
You can find a term of a sequence by repeatedly multiplying the first term by the common ratio.
Recognizing Growth Patterns of
Geometric Sequences in Context
Example 2

QUESTIONING STRATEGIES
If you know that a sequence is a geometric
sequence, how can you find the common
ratio? Choose any term after the first term and
divide it by the preceding term. The result is the
common ratio.
A bungee jumper jumps from a bridge. The table shows
the bungee jumper’s height above the ground at the top
of each bounce. The heights form a geometric sequence.
What is the bungee jumper’s height at the top of the 5th
bounce?
Bounce
Height (feet)
1
200
2
80
3
32
First bounce
200 ft
Find r.
Second bounce
80 ft
80 = 0.4 = r
_
200
Third bounce
32 ft
ƒ(1) = 200
ƒ(2) = 80
= 200(0.4) or 200(0.4)
1
ƒ(3) = 32
= 80(0.4)
= 200(0.4)(0.4)
= 200(0.4)
2
© Houghton Mifflin Harcourt Publishing Company
In each case, to get ƒ(n), you multiply 200 by the common ratio, 0.4, n -1 times. That is, you multiply
n-1
200 by (0.4) .
The jumper’s height on the 5th bounce is ƒ(5).
4
5- 1
Multiply 200 by (0.4)
= (0.4) .
4
200(0.4) = 200(0.0256)
= 5.12
The height of the jumper at the top of the 5th bounce is 5.12 feet.
Module 12
549
Lesson 1
LANGUAGE SUPPORT
A1_MTXESE353879_U5M12L1 549
Connect Vocabulary
Remind students that in a geometric sequence, the ratio of successive terms is
constant. This constant ratio is called the common ratio, often written as r. Point
out that it is called a common ratio because it is shared by all the pairs of successive
terms. Discuss other uses of this meaning of the word common; for example, a
common boys’ name.
549
Lesson 12.1
13/11/14 4:45 PM
Example 2

HOME CONNECTION
A ball is dropped from a height of 144 inches. Its height on the 1st bounce is 72 inches. On the 2nd
and 3rd bounces, the height of the ball is 36 inches and 18 inches, respectively. The heights form a
geometric sequence. What is the height of the ball on the 6th bounce to the nearest tenth of an inch?
Have students think of a real-world situation that can
be represented by a geometric sequence.
Find r.
36 = _
1 =r
_
72
2
ƒ(1) = 72
ƒ(2) = 36
__ ) ( __ )
= 72(
or 72
1
2
1
2
1
ƒ(3) = 18
( __ )
__ )( __ )
= 72(
__ )
= 72(
= 36
1
2
1
2
1
2
1
2
2
In each case, to get f(n), you multiply 72 by the common ratio,
multiply 72 by
(__12 )
2
, n - 1 times. That is, you
n-1
.
The height of the ball on the 6th bounce is f
5
6-1
1
1
2
2
Multiply 72 by
=
.
(_)
(_)
(
6
).
5
= 72(0.3125)
= 2.25
The height of the ball at the top of the 6th bounce is about
2.3
inches.
Reflect
7.
Is it possible for a sequence that describes the bounce height of a ball to have a common ratio greater
than 1?
No; if the common ratio were greater than 1, the bounce height would increase, which
© Houghton Mifflin Harcourt Publishing Company
1
(_12 ) = 72(_
32 )
72
__1
could not happen in the real world.
Module 12
A1_MTXESE353879_U5M12L1.indd 550
550
Lesson 1
2/19/14 3:31 AM
Understanding Geometric Sequences
550
Your Turn
ELABORATE
8.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Critical Thinking
Bounce
Use examples to help students make generalizations
about geometric sequences of positive numbers for
which the common ratio r is greater than 1 and
geometric sequences of positive numbers for which
the common ratio r is between 0 and 1,
that is, 0 < r < 1.
2
3
4
Term
1
2
4
8
Start with 1 and multiply
each term by 2 to get the
next term.
9.
Suppose all the terms of a geometric sequence are positive, and the common ratio r is between 0 and 1.
Is the sequence increasing or decreasing? Explain.
If r is between 0 and 1 and all the terms are positive, then each term is less than the
preceding term. So, the sequence is decreasing.
10. Essential Question Check-In If the common ratio of a geometric sequence is less than 0, what do you
know about the signs of the terms of the sequence? Explain.
The signs of the terms alternate. If r < 0 and the first term is negative, the second is
positive. Then the third must be negative. The signs of the terms continue to alternate.
If the first term is positive, the results are similar.
A1_MTXESE353879_U5M12L1.indd 551
Lesson 12.1
3
The ball bounces about 2.5 meters on the 4th bounce.
Module 12
551
3.375
= 2.53125
© Houghton Mifflin Harcourt Publishing Company
1
Words
3
Elaborate
Term
Number
n -1
4.5
f(4) = 6(0.75)
Ways to Represent the Geometric
Sequence 1, 2, 4, 8, …
f(n) = 1(2)
6
2
6
4-1
f(4) = 6(0.75)
Complete the graphic organizer with students to
discuss and summarize lesson concepts. In each box,
write a way to represent the geometric sequence.
Formula
Height (m)
1
4.5
_
= 0.75 = r
SUMMARIZE THE LESSON
Table
Physical Science A ball is dropped from a height of 8 meters. The table shows the height of each bounce.
The heights form a geometric sequence. How high does the ball bounce on the 4th bounce? Round your
answer to the nearest tenth of a meter.
551
Lesson 1
2/19/14 3:31 AM
Evaluate: Homework and Practice
EVALUATE
• Online Homework
• Hints and Help
• Extra Practice
Find the common ratio r for each geometric sequence and use r to
find the next three terms.
1.
3.
5, 15, 45, 135 …
15
=3=r
5
f(4) = 135, so f(5) = 135(3) = 405.
2.
_
-2, 6, -18, 54 …
6
= -3 = r
-2
f(4) = 54, so f(5) = 54(-3) = -162.
_
f(6) = 405(3) = 1215
f(6) = -162(-3) = 486
f(7) = 1215(3) = 3645
The next three terms of the sequence are
405, 1215, and 3645.
f(7) = 486(-3) = -1458
The next three terms of the sequence are
-162, 486, and -1458.
4, 20, 100, 500, …
4.
20
_
=5=r
4
f(4) = 500, so f(5) = 500(5) = 2500.
ASSIGNMENT GUIDE
8, 4, 2, 1, …
_4 = _1 = r
(_)
8
2
1
1
f(4) = 1, so f(5) = 1
= .
2
2
1 1
1
=
f(6) =
4
2 2
1 1
1
(
)
=
f 7 =
4 2
8
_(_) _
_(_) _
f(6) = 2500(5) = 12,500
f(7) = 12,500(5) = 62,500
The next three terms of the sequence are
2500, 12,500, and 62,500.
_
The next three terms of the sequence are
_1, _1, and _1.
2 4
5.
72, -36, 18, -9, …
-36
1 =r
_
= -_
6.
f(6) = 5.12(-0.4) = -2.048
f(7) = -2.048(-0.4) = 0.8192
The next three terms of the sequence are
5.12, -2.048, and 0.8192.
4.5, -2.25, and 1.125.
7.
10, 30, 90, 270, …
8.
30
_
=3=r
f(6) = 810(3) = 2430
Exercise
A1_MTXESE353879_U5M12L1 552
Exercise 22
Explore 2
Comparing Growth Patterns of
Arithmetic and Geometric
Sequences
Exercise 23
Example 1
Extending Geometric Sequences
Exercises 1–14, 21
Example 2
Recognizing Growth Patterns of
Geometric Sequences in Context
Exercises 15–20
_3 = 0.6 = r
f(6) = 0.648(0.6) = 0.3888
f(7) = 2430(3) = 7290
The next three terms of the sequence are
810, 2430, and 7290.
Module 12
5, 3, 1.8, 1.08, …
5
f(4) = 1.08, so f(5) = 1.08(0.6) = 0.648.
10
f(4) = 270, so f(5) = 270(3) = 810.
Explore 1
Exploring Growth Patterns of
Geometric Sequences
-80
_
= -0.4 = r
200
f(4) = -12.8, so f(5) = -12.8(-0.4) = 5.12.
72
2
f(4) = -9, so f(5) = -9 - 1 = 4.5.
2
f(6) = 4.5 - 1 = -2.25
2
f(7) = -2.25 - 1 = 1.125
2
The next three terms of the sequence are
( _)
( _)
200, -80, 32, -12.8, …
Practice
© Houghton Mifflin Harcourt Publishing Company
( _)
8
Concepts and Skills
f(7) = 0.3888(0.6) = 0.23328
The next three terms of the sequence are
0.648, 0.3888, and 0.23328.
Lesson 1
552
Depth of Knowledge (D.O.K.)
Mathematical Processes
1–10
1 Recall
1.D Multiple Representations
11–14
2 Skills/Concepts
1.C Select Tools
15–20
1 Recall
1.C Select Tools
21
2 Skills/Concepts
1.F Analyze Relationships
22
3 Strategic Thinking
1.C Select Tools
23
3 Strategic Thinking
1.G Explain and Justify Arguments
13/11/14 4:45 PM
Understanding Geometric Sequences
552
10. 243, 162, 108, 72, …
162
2
= =r
36
=2=r
243
3
18
2
f(4) = 72, so f(5) = 72
= 48.
f(4) = 144, so f(5) = 144(2) = 288
3
2
f(6) = 48
= 32
f(6) = 288(2) = 576
3
f(7) = 576(2) = 1152
2
1
= 21
f(7) = 32
3
3
The next three terms of the sequence are
The next three terms of the sequence are
288, 576, and 1152.
1
48, 32, and 21 .
3
Find the indicated term of each sequence by repeatedly multiplying the first term
by the common ratio. Use a calculator.
9.
AVOID COMMON ERRORS
18, 36, 72, 144
_ _
_
When finding a common ratio, students might divide
in the wrong order. Make sure that students divide
each term after the first term by the preceding term
to find the common ratio.
(_)
(_)
(_)
_
_
12. 16, -3.2, 0.64, …; 7th term
11. 1, 8, 64, …; 5th term
_
-3.2
_
= -0.2 = r
8
=8=r
1
5-1
f(5) = 1(8)
= 1(8)
16
7-1
f(7) = 16(-0.2)
4
= 16(-0.2)
= 4096
= 0.001024
13. -50, 15, -4.5, …; 5th term
14. 3, -12, 48, …; 6th term
_
-12
_
= -4 = r
15
= -0.3 = r
-50
5-1
f(5) = -50(-0.3)
= -50(-0.3)
6
3
6-1
f(6) = 3(-4)
= 3(-4)
4
= -0.405
5
= -3072
© Houghton Mifflin Harcourt Publishing Company
Solve. You may use a calculator and round your answer to the nearest tenth of
a unit if necessary.
15. Physical Science A ball is dropped from a height of 900 centimeters. The table
shows the height of each bounce. The heights form a geometric sequence. How high
does the ball bounce on the 5th bounce?
Bounce
Height (cm)
1
800
2
560
3
392
560
_
= 0.7 = r
800
Find the 5th term of the sequence.
f(5) = 800(0.7)
= 800(0.7)
5-1
4
= 192.08 centimeters
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16. Leo’s bank balances at the end of months 1, 2, and 3 are $1500, $1530, and $1560.60,
respectively. The balances form a geometric sequence. What will Leo’s balance be after
9 months?
The first two terms of the sequence are 1500 and 1530.
1530
= 1.02 = r
1500
Find the 9th term of the sequence.
CRITICAL THINKING
Have students explain why some sequences alternate
signs and some do not. (When the common ratio, r,
is negative, the terms in a geometric sequence
alternate signs. When the common ratio is positive,
the terms are all positive or all negative.)
_
f(9) = 1500(1.02)
= 1500(1.02)
9-1
8
= $1757.49
17. Biology A biologist studying ants started on day 1 with
a population of 1500 ants. On day 2, there were 3000 ants,
and on day 3, there were 6000 ants. The increase in an ant
population can be represented using a geometric sequence.
What is the ant population on day 5?
3000
_
=2=r
1500
Find the 5th term of the sequence.
f(5) = 1500 · 2 5-1
= 1500 · 2 4
= 24,000 ants
400
_
= 0.8 = r
Bounce
Height (cm)
1
500
2
400
3
320
500
Find the 8th term of the sequence.
f(8) = 500(0.8)
= 500(0.8)
8-1
7
= 104.8576 centimeters.
19. Finance The table shows the balance in an
investment account after each month. The balances
form a geometric sequence. What is the amount in
the account after month 6?
2040
= 1.2 = r
1700
Find the 6th term of the sequence.
_
f(6) = 1700(1.2)
= 1700(1.2)
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= $4230.14
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Month
Amount ($)
1
1700
2
2040
3
2448
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jolanta
Dabrowska/Alamy
18. Physical Science A ball is dropped from a height
of 625 centimeters. The table shows the height
of each bounce. The heights form a geometric
sequence. How high does the ball bounce on the 8th
bounce?
6-1
5
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Understanding Geometric Sequences
554
20. Biology A turtle population grows in a manner that can be
represented by a geometric sequence. Given the table of values,
determine the turtle population after 6 years.
JOURNAL
Have students determine whether each sequence is
arithmetic, geometric, or neither, and explain their
reasoning:
Year
3, 6, 9, 12, 15, … arithmetic
Number of Turtles
1
5
2
15
3
45
15
_
=3=r
3, 6, 10, 15, 21, … neither
5
Find the 6th term of the sequence.
3, 6, 12, 24, 48, … geometric
f(6) = 5 · 3 6-1 = 5 · 3 5 ; 1215 turtles
21. Consider the geometric sequence –8, 16, –32, ... Which of the following statements
are true?
a. The common ratio is 2.
16
___
= -2, so the common ratio is -2.
-8
b. The 5th term of the sequence is –128.
The common ratio is -2, and the 1st term is -8, f(n) = -8 (-2)
n-1
, and
f (5) = -8 (-2) = -128, and the 5th term of the sequence is -128.
4
c.
The 7th term is 4 times the 5th term.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Olha K/
Shutterstock
The 7th term is -2 times the 6th term, which is -2 times the 5th term, so the 7th term is 4
times the 5th term.
d. The 8th term is 1024.
The 8th term is -8 (-2) = 1024.
7
e. The 10th term is greater than the 9th term.
-8 (-2) is positive and -8 (-2) is negative, so the 10th term is greater than the 9th term.
9
So, statements b, c, d, and e are all true.
H.O.T. Focus on Higher Order Thinking
22. Justify Reasoning Suppose you are given a sequence with r < 0. What do you
know about the signs of the terms of the sequence? Explain.
Because r < 0, if the first term is negative, the second is positive. Then the
third must be negative. The signs of the terms continue to alternate.
23. Critique Reasoning Miguel writes the following: 8, x, 8, x, … He tells Alicia that
he has written a geometric sequence and asks her to identify the value of x. Alicia says
the value of x must be 8. Miguel says that Alicia is incorrect. Who is right? Explain.
The value that Alicia gave is correct, but it is not the only correct value.
x could also be -8. 8, -8, 8, -8, ... is also a geometric sequence, so
Miguel is correct.
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Lesson Performance Task
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
Multi-Step Gifford earns money by shoveling snow for the winter.
He offers two payment plans: either pay $400 for the entire winter
or pay $5 for the first week, $10 for the second week, $20 for the
third week, and so on. Explain why each plan does or does not form
a geometric sequence. Then determine the number of weeks after
which the total cost of the second plan will exceed the total cost of the
first plan.
Guide students who need help in representing the
geometric sequence 5, 10, 20, … by giving them an
explicit function.
Each plan forms a geometric sequence. In the first plan, the
geometric sequence is f(n) = 400. In the second plan,
n-1
the geometric sequence is g(n) = 5(2) .
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Reasoning
Create a table of values with columns listing the week numbers, the weekly payments, and
the total paid up to and including that week for the second plan.
In week 6, the second plan’s payment will be $160, and the total payments for weeks 1–6
will be $320, which is less than the $400 payment for the first plan. In week 7, the second
plan’s payment will be $320, and the total payments for weeks 1–7 will be $635, which
is more than the $400 payment for the first plan. So, after 7 weeks, the total cost of the
second plan will exceed the total cost of the first plan.
Have students explain how they could use reasoning
to check their answers for the approximate number of
weeks when the plans cost about the same amount of
money. For example, some students may say they
wrote out the geometric sequence until they could see
that $400 would occur between weeks 7 and 8.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Trudy
Wilkerson/Shutterstock
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Have students explain which of the following two options they would prefer. Have
students explain their reasoning.
13/11/14 10:26 PM
Option 1: They receive 1¢ on the first day, 2¢ on the second day, 4¢ on the third
day, 8¢ on the fourth day, and so on, doubling each day, for a total of 20 days.
Option 2: They receive $5000 on the first day and $0 after that.
Students will find that with Option 1, on the 20th day, without including the
amounts from the previous 19 days, the amount received would be $5242.88,
which is more than $5000. So, the total amount received in Option 1 would be
much more than the total amount, $5000, received in Option 2.
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.
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