Name
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Date
Additional Vocabulary Support
9-1
Mathematical Patterns
Choose the word or phrase from the list that best matches each sentence.
explicit formula
1. an ordered list of numbers
recursive formula
sequence
term
sequence
2. a formula that describes the nth term of a sequence using the number n
explicit formula
term
3. each number in a sequence
4. a formula that describes the nth term of a sequence by referring to preceding terms
recursive formula
Choose the word or phrase from the list that best completes each sentence.
explicit formula
initial condition
recursive formula
sequence
subscript number
term
5. For a sequence that is described by a recursive formula, the first term in the
sequence is the
initial condition
.
term
6. In the sequence 2, 4, 6, 8, the number 4 is the second
sequence.
7. The position of a term in a sequence can be represented by using
a(n) subscript number .
8. The formula an 5 3n 1 2 is a(n)
explicit formula
sequence
9. An ordered list of numbers is called a(n)
10. The formula an11 5 an 1 5 is a(n)
.
.
recursive formula .
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Think About a Plan
9-1
Mathematical Patterns
Geometry Suppose you are stacking boxes in levels that form squares. The
numbers of boxes in successive levels form a sequence. The figure at the right
shows the top four levels as viewed from above.
a. How many boxes of equal size would you need for the next lower level?
b. How many boxes of equal size would you need to add three levels?
c. Suppose you are stacking a total of 285 boxes. How many levels will you have?
1. How many boxes are in each of the first four levels?
1 z
Level 1: z 4 z
Level 2: z 9 z
Level 3: z 16 z
Level 4: z 2. How many boxes of equal size would you need for the next lower level? 25
3. What is a recursive or explicit formula that describes the number of boxes
in the nth level? explicit: an 5 n2
4. How many boxes would you need to add three levels?
25 z
z 1
36 z
z 1
49 z
z 5
z 110
z
5. What is a recursive or explicit formula that describes the total number
of boxes in a stack of n levels? recursive: a1 5 1; an 5 an21 1 n2
6. How can you use your formula to find the number of levels you will have
with a stack of 285 boxes?
Use the formula to find the number of boxes in a stack with successive levels until
an L 285
.
7. Suppose you are stacking a total of 285 boxes. Use your formula to find how
many levels you will have. Show your work.
a1 5 1, a2 5 1 1 22 5 5, a3 5 5 1 32 5 14, a4 5 14 1 42 5 30, a5 5 30 1 52 5 55,
a6 5 55 1 62 5 91, a7 5 91 1 72 5 140, a8 5 140 1 82 5 204, a9 5 204 1 92 5 285
z
z
8. You need levels to make a stack of 285 boxes.
9
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Practice
Form G
Mathematical Patterns
Find the first six terms of each sequence.
1. an 5 22n 1 1
21, 23, 25, 27, 29, 211
2. an 5 n2 2 1
0, 3, 8, 15, 24, 35
3. an 5 2n2 1 1
3, 9, 19, 33, 51, 73
4. an 5 1n 1 1
2, 2, 2, 2, 2, 2
5. an 5 2n 1 2
4, 6, 10, 18, 34, 66
6. an 5 2n2 2 n
1, 6, 15, 28, 45, 66
7. an 5 4n 1 n2
5, 12, 21, 32, 45, 60
1
8. an 5 3 n3
64 125
1 8
3 , 3 , 9, 3 , 3 , 72
9. an 5 (22)n
22, 4, 28, 16, 232, 64
Write a recursive definition for each sequence.
10. 214, 28, 22, 4, 10, c
an 5 an21 1 6 where
a1 5 214
13. 1, 3, 9, 27, c
an 5 3an21 where
a1 5 1
16. 36, 39, 42, 45, 48, c
an 5 an21 1 3 where
a1 5 36
11. 6, 5.7, 5.4, 5.1, 4.8, c
an 5 an21 2 0.3 where
a1 5 6
1 1 1 1
14. 1, 2, 4, 8, 16, c
12. 1, 22, 4, 28, 16, c
an 5 22an21 where
a1 5 1
2
1 2
15. 3 , 1, 1 3 , 1 3, 2, c
an 5 an21 1 13 where a1 5 23
an 5 12 an21 where a1 5 1
17. 36, 30, 24, 18, 12, c
an 5 an21 2 6 where
a1 5 36
18. 9.6, 4.8, 2.4, 1.2, 0.6, c
an 5 12 an21 where a1 5 9.6
19. 7, 14, 21, 28, 35, c
20. 2, 8, 14, 20, 26, c
21. 5, 6, 7, 8, 9, c
Write an explicit formula for each sequence. Find the twentieth term.
an 5 7n; 140
an 5 6n 2 4; 116
an 5 n 1 4; 24
22. 21, 0, 1, 2, 3, c
an 5 n 2 2; 18
23. 3, 5, 7, 9, 11, c
an 5 2n 1 1; 41
24. 0.8, 1.6, 2.4, 3.2, 4, c
an 5 0.8n; 16
5
1 1 3
25. 4, 2, 4, 1, 4, c
1 1 1 1 1
26. 2, 4, 6, 8, 10, c
2 2 2 2 2
27. 3, 13, 23, 33, 43, c
1 1
; 40
an 5 2n
an 5 n 2 13; 19 23
an 5 n4 ; 5
Find the eighth term of each sequence.
28. 1, 3, 5, 7, 9, c
15
29. 400, 200, 100, 50, 25, c
3.125
30. 0, 22, 24, 26, 28, c
214
31. 1, 2, 4, 8, 16, c
128
32. 44, 39, 34, 29, 24, c
9
1 1
35. 14, 2 2, 5, 10, 20, c
160
33. 0.7, 0.8, 0.9, 1.0, 1.1, c
1.4
34. 4, 11, 18, 25, 32, c
c
53
36. 26, 29, 212, 215, 218,
227
37. A man swims 1.5 mi on Monday, 1.6 mi on Tuesday, 1.8 mi on Wednesday,
2.1 mi on Thursday, and 2.5 mi on Friday. If the pattern continues, how many
miles will he swim on Saturday? 3.0 mi
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Practice (continued)
Form G
Mathematical Patterns
Determine whether each formula is explicit or recursive. Then find the first five
terms of each sequence.
1
38. an 5 3 n
explicit; 13, 23, 1, 43, 53
39. an 5 n2 2 6
explicit; 25, 22, 3, 10, 19
40. a1 5 5, an 5 3an21 2 7
recursive; 5, 8, 17, 44, 125
1
41. an 5 2 (n 2 1)
explicit; 0, 12, 1, 1 12, 2
42. a1 5 5, an 5 3 2 an21
recursive; 5, 22, 5, 22, 5
43. a1 5 24, an 5 2an21
recursive; 24, 28, 216, 232, 264
44. Error Analysis Your friend says the explicit formula for the sequence
1, 8, 27, 64 is an 5 n2. Is she correct? Explain. She is incorrect; in order to find each term
in the sequence, the term number must be cubed, not squared.
45. Writing Explain how to find an explicit formula for a sequence. Look for a pattern in the
sequence and find a mathematical rule that gives the nth term, given the number n.
46. The first figure of a fractal contains one segment. For each successive figure,
six segments replace each segment.
a. How many segments are in each of the first four figures of the sequence? 1, 6, 36, 216
b. Write a recursive definition for the sequence. an 5 6an21 where a1 5 1
47. The sum of the measures of the exterior angles of any polygon is 3608. All the
angles have the same measure in a regular polygon.
a. Find the measure of one exterior angle in a regular hexagon (six angles). 608
b. Write an explicit formula for the measure of one exterior angle in a regular
polygon with n angles. an 5 360
n
c. Why would this formula not be meaningful for n 5 1 or n 5 2?
No polygon has one or two angles.
48. Reasoning In order to find a term in a sequence, its position in the sequence is doubled
and then two is added. What are the first ten terms in the sequence?
4, 6, 8, 10, 12, 14, 16, 18, 20, 22
49. Writing Explain the difference between a recursive and an explicit formula.
An explicit formula defines how to find the nth term directly from the number n, while
a recursive formula defines how to find each term from the previous term(s).
50. Open-Ended Write five terms in a sequence. Describe the sequence using a
recursive or explicit formula. Check students’ work.
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Practice
Form K
Mathematical Patterns
Find the first five terms of each sequence.
2. an 5 n2 1 4
5, 8, 13, 20, 29
1. an 5 4n 2 1
Substitute 1 for n and simplify.
a1 5 4(1) 2 1 5 3
Substitute 2 for n and simplify.
a2 5 4(2) 2 1 5 7
Continue for the numbers 3, 4, and 5.
The first five terms are 3, 7, z 11
z , z 15
z , and z 19
z .
1
3. a 5 2 n 1 2
2.5, 3, 3.5, 4, 4.5
4. an 5 3n
5. an 5 26n2
26, 224, 254, 296, 2150
3, 9, 27, 81, 243
6. Write an explicit formula for a sequence with 3, 5, 7, 9, and 11 as its first five terms.
an 5 2n 1 1
Write a recursive definition for each sequence.
7. 2, 6, 12, 20, c
8. 120, 60, 30, 15, c
a1 5 120; an 5 Q 12 R an21
Identify the initial condition.
a1 5 2
Use n to express the relationship between
successive terms.
an 5 an21 1 2n
9. 3, 8, 13, 18, c
a1 5 3; an 5 an21 1 5
10. 1, 3, 9, 27, c
a1 5 1; an 5 3an21
11. 2, 3, 8, 63, c
2
a1 5 2; an 5 an21
21
12. Writing Explain the difference between a recursive definition and an explicit
formula. An explicit formula describes the nth term of a sequence using the number n.
A recursive formula defines a sequence by the relationship between successive terms.
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Practice (continued)
Form K
Mathematical Patterns
Write an explicit formula for each sequence. Then find the tenth term.
13. 7, 10, 13, 16, c
an 5 3n 1 4
14. 8, 9, 10, 11, 12, c
an 5 n 1 7; 17
1
1
1
15. 2 2 , 0, 2 , 1, 1 2 , c
17. 3, 1, 21, 23, 25, c
an 5 22n 1 5; 215
18. 1, 7, 25, 79, 241
an 5 3n 2 2; 59,047
a10 5 3(10) 1 4 5 z 34
z
16. 1, 4, 9, 16, c
an 5 n2; 100
an 5 12n 2 1; 4
19. Reasoning You and your friend are trying to find the 80th term in the
sequence 8, 14, 20, 26, 32, c. You use a recursive definition and your friend
uses an explicit formula. Who will find the 80th term first? Why?
Your friend will find the 80th term first because he is using an explicit formula.
Your friend will substitute 80 into the formula to get the answer, while you will go
through 79 iterations of the recursive formula.
20. Your neighbor recently began learning to play the guitar. On the first day, she
practiced for 0.4 h. On the second day, she practiced for 0.5 h. She practiced
for 0.65 h on the third day, and 0.85 h on the fourth day. If this pattern
continues, how long will she practice on the seventh day? 1.75 h
21. Charles lost two rented movies, so he owes the rental store a fee of $40. At the
end of each month, the amount that Charles owes will increase by 5%, plus
a $2 billing fee. How much money will Charles owe the rental store after
8 months? $78.20
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Standardized Test Prep
Mathematical Patterns
Multiple Choice
For Exercises 1−6, choose the correct letter.
1. What are the first five terms of the sequence? C
an 5 3n 2 1
2, 5, 8, 11, 14
2, 8, 26, 80, 242
3, 9, 27, 81, 243
2, 4, 8, 16, 32
2. The formula an 5 3n 1 2 best represents which sequence? G
3, 6, 9, 12, 15
4, 7, 10, 13, 16
5, 8, 11, 14, 17
5, 9, 29, 83, 245
3. Which pattern can be represented by an 5 n2 2 3? D
21, 0, 5, 12, 21
4, 7, 12, 19, 28
1, 4, 9, 16, 25
22, 1, 6, 13, 22
4. The sequence 4, 16, 36, 64, 100, ccan best be represented by which formula? G
an 5 4n
an 5 4n2
an 5 4n3
an 5 2n4
5. For the sequence 0, 6, 16, 30, 48, c, what is the 40th term? A
3198
3200
4000
16,000
6. A student sets up a savings plan to transfer money from his checking account
to his savings account. The first week $10 is transferred, the second week $12
is transferred, the third week $16 is transferred, and the fourth week $24 is
transferred. If this pattern continues and he starts with $100 in his checking
account, how many weeks will pass before his balance is zero? G
4
5
6
7
Short Response
7. After training for and running a marathon, an athlete wants to reduce her daily run
by half each day. The marathon is about 26 mi. How many days will it take after the
marathon before she runs less than a mile a day? Show your work.
[2] 5 days; Day 1: 13 mi, Day 2: 6.5 mi, Day 3: 3.25 mi, Day 4: 1.625 mi, Day 5: 0.8125 mi
[1] correct answer, without work shown OR incorrect answer with correct sequence
[0] incorrect answers and no work shown OR no answers given
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Enrichment
Mathematical Patterns
You can define the terms in a sequence using an explicit formula or a recursive
definition. You can use another method, called iteration, to form a sequence. The word
iteration means to repeat an action. In mathematics, a sequence of numbers is generated
through iteration when the same procedure is performed on each output.
1. Consider the function f (x) 5 5x 1 1. Let the first term of a sequence be 0.
What is f (0)? Let f (0) be the second term of the sequence. Write the sequence. 0, 1
2. To create more terms of this sequence through iteration, continue to apply
f (x) to each output. The third term in this sequence can be described as
f ( f (0)). What is the third term? f (f (0)) 5 f (1) 5 6
3. Determine the first 10 terms of this sequence. You already have the first 3 terms.
0; 1; 6; 31; 156; 781; 3906; 19,531; 97,656; 488,281
4. Determine the first 5 terms of the sequence formed through iterations of
x
f (x) 5 2 1 1. Begin with x 5 2. Describe the sequence.
2, 2, 2, 2, 2; all of the terms in the sequence are 2.
5. Will you get the same type of sequence if you start with a different number? No; for
example, if you start with x 5 0, the sequence is 0, 1, 1.5, c
6. Iterations have uses other than to form numerical sequences. Consider this iterative
process, which forms a sequence of a set of three integers. Make a set of any three
integers. Compute the absolute value of the difference between each pair of integers in
the set. This produces a new set of three integers. Continue this process on each new set
of three integers. Describe what eventually happens. No matter what three integers you
choose to start with, the set will eventually repeat itself in combinations of the set {0, a, a},
where a is a positive integer.
7. You can form fractals through iterations. Fractals are geometric figures just like circles
or rectangles, but fractals have a special property that these geometric figures do
not. You make fractals by iterating the figure itself. For example, start by drawing an
equilateral triangle on graph paper. Divide each side into three equal parts. Draw
another equilateral triangle on one side of the triangle that has the middle section as its
base. Repeat this process on the remaining two sides. You have just created the first two
iterations of a fractal called the Koch snowflake.
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Reteaching
Mathematical Patterns
Some patterns are much easier to determine than others. Here are some tips that
can help with unfamiliar patterns.
• If the terms become progressively smaller, subtraction or division may be
involved.
• If the terms become progressively larger, addition or multiplication may be
involved.
Problem
What is the next term in the sequence 6, 8, 11, 15, 20, c?
6
8
12
11
13
15
14
20
15
In each term, the number that is added
to the previous term increases by one.
Spread the numbers in the sequence apart, leaving
space between numbers.
Beneath each space, write what can be done to
get the next number in the sequence.
Find a pattern.
If the pattern is continued, the next term is 20 1 6, or 26.
Exercises
Describe the pattern that is formed. Find the next three terms.
1.
1.
2.
3.
5, 6, 8, 11, 15
4.
4.
5.
6.
1, 3, 9, 27, 81
7.
7.
8.
9.
5, 25, 125, 625, 3125
10.
10.
11.
12.
2. 3, 6, 12, 24, 48
3. 1, 22, 4, 28, 16, 232
Each term is increased by one more than the previous term; 20, 26, 33
Each term is multiplied by 2 to get the next term; 96, 192, 384
Each term is multiplied by 22 to get the next term; 64, 2128, 256
5. 100, 95, 90, 85, 80
6. 15, 18, 21, 24, 27
Each term is multiplied by 3 to get the next term; 243, 729, 2187
Each term is decreased by 5 to get the next term; 75, 70, 65
Each term is increased by 3 to get to the next term; 30, 33, 36
8. 50, 49, 47, 44, 40
9. 240, 120, 60, 30, 15
Each term is multiplied by 5 to get the next term; 15,625; 78,125; 390,625
Each term is decreased by one more than the previous term; 35, 29, 22
Each term is divided by 2 to get the next term; 7.5, 3.75, 1.875
3, 5, 9, 15, 23
11. 280, 120, 2180, 270, 2405 12. 1, 5, 13, 29, 61
In each term, the number is increased by two more than the previous term; 33, 45, 59
Each term is multiplied by 21.5 to get the next term; 607.5, 2911.25, 1366.875
Each term is multiplied by 2 and then 3 is added to get the next term; 125, 253, 509
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Reteaching (continued)
9-1
Mathematical Patterns
To find a recursive definition for a sequence, you compare each term to the
previous term.
Problem
What is the recursive definition for the sequence?
800, 2400, 200, 2100, 50, c
To find the recursive definition for a sequence, first describe the sequence in words.
The initial term is 800.
The terms are alternatively negative g
g and positive.
800,
200,
2400,
2100,
50, g
To find the next term in the sequence,
divide by negative two.
The next term in the
sequence will be 225.
Now translate the description into the parts of the recursive formula.
a1 5 800
The initial term is 800.
an 5 an21 4 (22)
To find the next term, divide the previous term by 22.
Exercises
Write a recursive definition for each sequence.
13. 38, 33, 28, 23, c
an 5 an21 2 5 where
a1 5 38
14. 7, 14, 28, 56, c
an 5 2an21 where a1 5 7
16. 2, 6, 18, 54, c
17. 4.5, 5, 5.5, 6, c
an 5 3an21 where a1 5 2
an 5 an21 1 0.5 where
a1 5 4.5
15. 25, 27, 29, 211, c
an 5 an21 2 2 where
a1 5 25
18. 17, 20, 24, 29, c
an 5 an21 1 (n 1 1)
where a1 5 17
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Additional Vocabulary Support
Arithmetic Sequences
Arithmetic Sequence
An arithmetic sequence is a sequence where the difference between consecutive terms is
constant.
a, a 1 d, a 1 2d, a 1 3d, c
Sample
2, 5, 8, 11, 14, c
Determine whether or not each sequence is arithmetic.
1. 1, 4, 7, 9, 11, c
not arithmetic
2. 3, 9, 15, 21, 27, c
arithmetic
3. 0, 15, 30, 45, 60, c
arithmetic
4. 0, 1, 3, 6, 10, c
not arithmetic
Use the formula an 5 a 1 (n 2 1)d to find the indicated term in each
arithmetic sequence.
5. Find the 12th term in the sequence that begins 3, 6, 9, c
36
6. Find the 38th term in the sequence that begins 4, 10, 16, c
226
7. Find the 104th term in the sequence that begins 5, 9, 13, c
417
Arithmetic Mean
The arithmetic mean is the average of a set of numbers. The arithmetic mean of two
numbers x and y is found using the formula displayed below.
x1y
2
Sample
The arithmetic mean of 4 and 6 is
416
10
2 5 2 5 5.
Find the missing number in the arithmetic sequence. This number is the
arithmetic mean of the two given numbers.
8. c, 13,
, 37, c
25
9. c, 26,
, 42, c
34
10. c, 45,
, 99, c
72
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Arithmetic Sequences
Transportation Suppose a trolley stops at a certain intersection every 14 min. The
first trolley of the day gets to the stop at 6:43 a.m. How long do you have to wait for
a trolley if you get to the stop at 8:15 a.m.? At 3:20 p.m.?
Know
z
z
1. If you define 12:00 a.m. as minute 0, then 6:43 a.m. is 403
min from 0.
z
z
z
z
2. 8:15 a.m. is 495
min from 0 and 3:20 p.m. is 920
min from 0.
z
z
3. The trolley stops every 14 min .
Need
4. To solve the problem I need to find:
the closest times that the trolley gets to the stop that are after 8:15 A.M.
and 3:20 P.M.
.
Plan
5. What is an explicit formula for the number of minutes after 12:00 a.m. that the
trolley gets to the stop?
an 5 403 1 (n 2 1)14
6. Use your formula to find the smallest n that gives the minutes just after
8:15 a.m. that the trolley arrives at the stop. 8
7. Using this n in your formula, when does the trolley stop? at 501 min
How long do you have to wait for this trolley? 6 min
8. Use your formula to find the smallest n that gives the minutes just after 3:20 p.m.
that the trolley arrives at the stop. 38
9. Using this n in your formula, when does the trolley stop? at 921 min
How long do you have to wait for this trolley? 1 min
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Form G
Arithmetic Sequences
Determine whether each sequence is arithmetic. If so, identify the common
difference.
1. 2, 3, 5, 8, c no
2. 0, 23, 26, 29, c yes; 23
3. 0.9, 0.5, 0.1, 20.3, c yes; 20.4
4. 3, 8, 13, 18, . . . yes; 5
5. 14, 215, 244, 273, c yes; 229
6. 3.2, 3.5, 3.8, 4.1, c yes; 0.3
7. 234, 228, 222, 216, c yes; 6
8. 2.3, 2.5, 2.7, 2.9, c yes; 0.2
9. 127, 140, 153, 166, c yes; 13
10. 11, 13, 17, 25, c no
Find the 43rd term of each sequence.
11. 12, 14, 16, 18, c 96
12. 13.1, 3.1, 26.9, 216.9, c 2406.9
13. 19.5, 19.9, 20.3, 20.7, c 36.3
14. 27, 24, 21, 18, c 299
15. 2, 13, 24, 35, c 464
16. 21, 15, 9, 3, . . . 2231
17. 1.3, 1.4, 1.5, 1.6, c 5.5
18. 22.1, 22.3, 22.5, 22.7, c 210.5
19. 45, 48, 51, 54, c 171
20. 20.073, 20.081, 20.089, c 20.409
Find the missing term of each arithmetic sequence.
21. c 23, 7 , 49, c 36
22. 14, 7 , 28, c 21
24. c 14, 7 , 15, c 14.5
25. c 245, 7 , 239, c 242 26. c 25, 7 , 22, c −3.5
27. 22, 7 , 2, c 0
28. c 26, 7 , 2, c 22
30. c 245, 7 , 212, c 228.5 31. 22, 7 , 456, c 227
23. c 29, 7 , 33, c 31
29. 234, 7 , 77, c 21.5
32. c 34, 7 , 345, c 189.5
33. A teacher 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 money to
protect 12 acres. How many acres will the teacher’s donations protect at the
end of the tenth year? 40 acres
34. Writing Explain how you know that the sequence 109, 105, 101, 97, 93, cis
arithmetic. The sequence has a common difference between terms of 24.
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Practice (continued)
Form G
Arithmetic Sequences
Find the arithmetic mean an of the given terms.
35. an21 5 5, an11 5 11 8
36. an21 5 17, an11 5 3 10
37. an21 5 28, an11 5 29 28.5
38. an21 5 20.6, an11 5 3.8 1.6
39. an21 5 y 2 z, an11 5 y y 2 2z
40. an21 5 2t 1 3, an11 5 4t 2 1 3t 1 1
41. Open-Ended Write an arithmetic sequence of at least five terms with a
positive common difference. a five-term sequence with a positive common
difference
42. Error Analysis On your homework, you write that the missing term in the
arithmetic sequence 31, ___, 41, c is 35 12 . Your friend says the missing term
is 36. Who is correct? What mistake was made? Your friend is correct. You did not take
the average of 31 and 41 correctly to find the missing term of 36.
43. Reasoning Explain why 84 is the missing term in the sequence 89, 86.5, ___, 81.5, c.
The common difference in the arithmetic sequence is 22.5, which means the missing term
must be 84 as that is 2.5 less than the term before it and 2.5 more than the term after it.
44. Writing Describe the general process of finding a missing term in an
arithmetic sequence. If the term that is missing occurs between two other terms that
are consecutive to the missing term, you can take the arithmetic mean of the two terms. If
the term that is missing is not consecutive, use the formula an 5 a 1 (n 2 1)d.
45. You are making an arrangement of cubes in concentric rings for a sculpture.
The number of cubes in each ring follows the pattern below.
1, 9, 17, 25, 33, c
a. Is this an arithmetic sequence? Explain. Yes; there is a common difference of 8.
b. What are the next three terms? 41, 49, 57
c. If the sequence continues to the 100th term in this pattern, what will that term be? 793
46. Each year, a volunteer organization expects to add 5 more people to the
number of shut-ins for whom the group provides home maintenance services.
This year, the organization provides the service for 32 people.
a. Write a recursive formula for the number of people the organization
expects to serve each year. an 5 an21 1 5 where a1 5 32
b. Write the first five terms of the sequence. 32, 37, 42, 47, 52
c. Write an explicit formula for the number of people the organization expects
to serve each year. an 5 32 1 5(n 2 1)
d. How many people would the organization expect to serve in the 20th year? 127 people
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Name
9-2
Class
Date
Practice
Form K
Arithmetic Sequences
Determine whether each sequence is arithmetic. If so, identify the common
difference.
1. 1, 4, 7, 10, c
2. 6, 10, 14, 18, 22, c
arithmetic; 4
42153
3. 1, 3, 6, 10, 15, c
not arithmetic
72453
10 2 7 5 3
This sequence is arithmetic.
3 z .
The common difference is z 4. 216, 213, 29, 24, 2, c
not arithmetic
5. 2, 9, 16, 23, 30, c
arithmetic; 7
6. 43, 56, 69, 82, c
arithmetic; 13
7. Reasoning Is the sequence represented by the formula an 5 4n 1 8
arithmetic? Explain.
Yes; the difference between consecutive terms is 4.
Find the 24th term of each arithmetic sequence.
8. 4, 6, 8, 10, 12, c
9. 2, 5, 8, 11, 14, c
an 5 a1 1 (n 2 1)d
an 5 a1 1 (n 2 1)d
a24 5 4 1 (24 2 1)2
71
10. 9, 5, 1, 23, 27, c
283
a24 5 4 1 46
a24 5 z 50
z
Find the missing terms in the following arithmetic sequences.
11. 2, ___, ___, 14, c
z
zz
z
9 , 15
12. 3, , 21, c
z
zz
z
13. 65, 54
, 43
, 32, c
14 5 2 1 3d
12 5 3d
d54
6 z
2 1 4 5 z 6 1 4 5 z 10
z
14. Error Analysis Noah used the formula an 5 a 1 (n 2 1)d to find the 12th
term in the sequence 2, 4, 7, 11, 16, c. Did Noah find the correct term? How
do you know? No; Noah applied the explicit formula for arithmetic sequences to a
sequence that is not arithmetic.
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Practice (continued)
9-2
Form K
Arithmetic Sequences
Find the missing term of each arithmetic sequence.
z
z
23 , 37, c
16. c9, 15. c4, ___, 18, c
Find the arithmetic mean of the given terms.
4 1 18 5 22
22 4 2 5 11
z
z
11 .
The missing term is z
z
17. 46, 37
, 28, c
z
z
18. 212, 28
, 24, c
z
z
19. c4, 220 , 244, c
20. Error Analysis Your friend used the arithmetic mean to find the missing term
in the following sequence: 3, ___, 29, 42, c. His answer was 13. What error
did your friend make? What is the correct answer?
He subtracted 3 from 29 when he should have added 3 and 29; 16
21. An architect is designing a building with sides in the shape of a trapezoid. The
number of windows on each floor forms an arithmetic sequence. There are
124 windows on the first floor and 116 windows on the second floor.
a. Write an explicit formula to represent the sequence. an 5 132 2 8n
b. How many windows are on the tenth floor? 52 windows
22. Your cousin opened a bank account with a deposit of $256 dollars. After one
week, she had $280 in her account. After two weeks, she had $304, and after
three weeks she had $328. If this pattern continues, how much money will
your cousin have in her account after 18 weeks? $688
23. There is a puddle 1.4 cm deep in your backyard. After one minute of rain, the
puddle was 1.45 cm deep. The puddle was 1.5 cm deep after it rained for two
minutes. If the pattern continues, how deep will the puddle be after it rains for
45 min? 3.65 cm
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Standardized Test Prep
Arithmetic Sequences
Multiple Choice
For Exercises 1−6, choose the correct letter.
1. Which sequence is an arithmetic sequence? A
7, 10, 13, 16, 19, c
7, 14, 28, 56, 112, c
7, 8, 10, 13, 17, c
1, 7, 14, 22, 31, 41, c
2. An arithmetic sequence begins 4, 9, c. What is the 20th term? I
76
80
84
99
3. What are the missing terms of the arithmetic sequence 5, __, __, 62, c? C
19, 24
19, 34
24, 43
43, 62
4. What is the missing term of the arithmetic sequence 25, __, 45, c? G
30
35
37
40
5. The seventh and ninth terms of an arithmetic sequence are 197 and 173. What
is the eighth term? C
161
180
185
221
6. An artist is creating a tile mosaic. She uses 4 green tiles in the first row, 11 green
tiles in the second row, 18 green tiles in the third row, and 25 green tiles in the
fourth row. If she continues the pattern, how many green tiles will she use in
the 20th row? I
32
58
134
137
Extended Response
7. What is the 100th term in the arithmetic sequence beginning with 3, 19, c?
Show your work.
[4] 1587; a 5 3, n 5 100, d 5 16, an 5 a 1 (n 2 1)d;
a100 5 3 1 (100 2 1)16 5 3 1 1584 5 1587
[3] appropriate method shown, with one computational error
[2] appropriate method shown, with several computational errors OR correct term found
incorrectly with work shown
[1] incorrect term, without work shown
[0] incorrect answers and no work shown OR no answers given
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Enrichment
9-2
Arithmetic Sequences
There are many types of sequences. One interesting type of sequence is the Farey
sequence. The first four Farey sequences are:
0
F1: e 1 , 11 f
0
F2: e 1 , 12 , 11 f
0
F3: e 1 , 13 , 12 , 23 , 11 f
0
3
F4: e 1 , 14 , 13 , 12 , 23 , 4 , 11 f
Each Farey sequence is a list of fractions in increasing order between 0 and 1,
written in simplest form with a denominator less than or equal to the integer n.
For any n greater than 1, there are an odd number of terms in the sequence and
the middle term is 12 .
Problem
What are the terms of the Farey sequence for n 5 5?
The Farey sequence for n 5 5 contains all the terms of the Farey sequence F4 plus
the fractions between 0 and 1 which have a denominator of 5 when written in
simplest form.
0
5
0
The fractions 5 and 5 will not be added because they simplify to 1 and 11 . Insert the
1 2 3
4
fractions 5 , 5 , 5 , and 5 in the Farey sequence F4.
0
3
3
F5: e 1 , 15 , 14 , 13 , 25 , 12 , 5 , 23 , 4 , 45 , 11 f
Exercises
1. How many terms are in each of the first five Farey sequences? 2, 3, 5, 7, 11
2. What are the terms for the Farey sequence F6? e 01, 16, 15, 14, 13, 25, 12, 35, 23, 34, 45, 56, 11 f
3. What will be the new terms in the Farey sequence F7? e 17, 27, 37, 47, 57, and 67 f
4. Since 11 is a prime number, how many more terms will be in the sequence F11
compared to the sequence F10? 10
5. Is there any limit to how large n can be? No, n can be any positive integer although the
computations become tedious.
6. Can you give examples of any other sequences?
Answers may vary. Sample: arithmetic, geometric, and Fibonacci
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Reteaching
Arithmetic Sequences
The explicit formula for the nth term of an arithmetic sequence is
an 5 a 1 (n 2 1)d.
• a is the starting value and d is the common difference.
• n is always greater than or equal to 1.
• You can write the sequence as a, a 1 d, a 1 2d, a 1 3d, c
Problem
Find the 15th term of an arithmetic sequence whose first three terms are
20, 16.5, and 13.
20 2 16.5 5 3.5
16.5 2 13 5 3.5
First, find the common difference. The difference between
consecutive terms is 3.5. The sequence decreases. The common
difference is 23.5.
an 5 a 1 (n 2 1) d
a15 5 20 1 (15 2 1)(23.5)
Use the explicit formula.
Substitute a 5 20, n 5 15, and d 5 23.5.
5 20 1 (14)(23.5)
Subtract within parentheses.
5 20 1 249
Multiply.
5 229
The 15th term is 229.
Check the answer. Write a1, a2, c, a15 down the left side of your paper. Start with
a1 5 20. Subtract 3.5 and record 16.5 next to a2. Continue until you find a15.
Exercises
Find the 25th term of each sequence.
1. 20, 18, 16, 14, c 228
2. 0.0057, 0.0060, 0.0063, c 0.0129
3. 4, 0, 24, 28, c 292
4. 0.2, 0.7, 1.2, 1.7, c 12.2
5. −10, 28.8, 27.6, 26.4, c 18.8
6. 22, 26, 30, 34, c 118
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Reteaching (continued)
Arithmetic Sequences
To solve word problems that involve arithmetic sequences, identify the common
difference d, the starting value a, and the number of terms in the sequence n.
Problem
As a part-time home health care aide, you are paid a weekly salary plus a fixed fuel
fee for every patient you visit. You receive $240 in a week that you visit 1 patient.
You receive $250 in a week that you visit 2 patients. How much will you receive if
you visit 12 patients in 1 week?
d 5 a2 2 a1 5 250 2 240 5 10
The common difference is the difference between two consecutive
terms. You receive $10 per visit.
a 5 240
Identify the starting value. You receive $240 for a week with 1 visit.
n 5 12
You want to find the earnings in a week in which you visit 12 patients.
an 5 a 1 (n 2 1)d
Write the formula for the nth term.
5 240 1 (12 2 1)10
Substitute.
5 240 1 110 5 350
Simplify.
You will earn $350 if you visit 12 patients in 1 week.
Exercises
7. Suppose you begin to work selling ads for a newspaper. You will be paid $50/wk
plus a minimum of $7.50 for each potential customer you contact. What is the least
amount of money you earn after contacting eight businesses in 1 wk? $110
8. A boy starts a savings account for a mountain bike. He initially deposits $15.
He decides to increase each deposit by $8. How much is his 17th deposit? $143
9. A woman is knitting a blanket for her infant niece. Each day, she knits four
more rows than the day before. She knitted seven rows on Sunday. How many
rows will she knit on the following Saturday? 31 rows
10. Joe started a 30-min workout program this week. He wants to increase the
workout by 5 min every week. How long will his program be in the 16th week? 105 min
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Additional Vocabulary Support
Geometric Sequences
Use the chart below to review vocabulary. These vocabulary words will help you
complete this page.
Vocabulary
Words
Explanations
Examples
Geometric
sequence
A sequence in which the ratio
of any term (after the first) to its
preceding term is a constant value.
The sequence 3, 6, 12, 24, . . .
is geometric because all of the
consecutive terms have a ratio of 2.
Common ratio
the ratio of each term to its
preceding term in a geometric
sequence
The common ratio in the
sequence 1, 4, 16, 64,
256, . . . is 4.
Geometric mean
The geometric mean of two
numbers x and y is !xy .
The geometric mean of
the numbers 4 and 9 is
!4 ? 9 5 !36 5 6.
1. The terms in the sequence 2, 6, 18, 54, 162, . . . all share a
common ratio
with their preceding terms.
2. The numbers 8 and 2 have a
geometric mean
of 4.
3. The consecutive terms in a geometric sequence all share a common ratio.
Identify each sequence as arithmetic or geometric.
4. 2, 8, 32, 128, . . .
geometric
5. 1, 3, 9, 27, . . .
geometric
6. 1, 4, 7, 10, . . .
arithmetic
Identify the common ratio for each geometric sequence.
7. 3, 12, 48, 192, . . .
4
8. 12, 60, 300, 1500, . . .
5
Find the missing term in the geometric sequence.
9. . . . , 4, ___, 16, . . .
8
10. . . . , 9, ___, 25, . . .
15
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Think About a Plan
Geometric Sequences
Athletics During your first week of training for a marathon, you run a total of
10 miles. You increase the distance you run each week by twenty percent. How many
miles do you run during your twelfth week of training?
Understanding the Problem
1. How can you write a sequence of numbers to represent this situation?
Answers may vary. Sample: Start with 10, and multiply it and each successive term
by 120% or 1.2
.
2. Is the sequence arithmetic, geometric, or neither?
geometric
3. What is the first term of the sequence? 10
4. What is the common ratio of the sequence? 1.2
5. What is the problem asking you to determine?
the 12th term of a geometric sequence that represents the number of miles you run
each week
Planning the Solution
6. Write a formula for the sequence.
an 5 10(1.2)n21
Getting an Answer
7. Evaluate your formula to find the number of miles you run during your
twelfth week of training.
about 74.3 miles
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Practice
Form G
Geometric Sequences
Determine whether each sequence is geometric. If so, find the common ratio.
1. 3, 9, 27, 81, . . . yes; 3
2. 4, 8, 16, 32, . . . yes; 2
3. 4, 8, 12, 16, . . . no
4. 4, 28, 16, 232, . . . yes; 22
5. 1, 0.5, 0.25, 0.125, . . . yes; 0.5 6. 100, 30, 9, 2.7, . . . yes; 0.3
7. 25, 0, 5, 10, . . . no
8. 64, 232, 16, 28, . . . yes; 20.59. 1, 4, 9, 16, . . . no
Find the tenth term of each geometric sequence.
10. 2, 4, 8, . . . 1024
11. 1, 3, 9, . . . 19,683
13. 23, 9, 227, . . . 59,049
14. 23, 212, 248, . . . 2786,43215. 25, 25, 2125, . . . 9,765,625
1 1 1
1
16. 3, 9, 27 , . . . 59,049
17. 0.3, 0.6, 1.2, . . . 153.6
12. 22, 6, 218, . . . 39,366
1 1
18. 4 , 2 , 1, . . . 128
19. When a pendulum swings freely, the length of its arc decreases geometrically.
Find each missing arc length.
a. 20th arc is 20 in.; 22nd arc is 18.5 in. about 19.2 in.
b. 8th arc is 27 mm; 10th arc is 3 mm 9 mm
c. 5th arc is 25 cm; 7th arc is 1 cm 5 cm
d. 100th arc is 18 ft; 98th arc is 2 ft 6 ft
Find the missing term of each geometric sequence. It could be the geometric
mean or its opposite.
20. 4, j , 16, . . . 68
21. 9, j , 16, . . . 612
22. 2, j , 8, . . . 64
23. 3, j , 12, . . . 66
24. 2, j , 50, . . . 610
25. 4, j , 5.76, . . . 64.8
26. 625, j , 25, . . . 6125
1
27. 3 , j , 3, . . . 61
28. 0.5, j , 0.125, . . . 60.25
29. Writing Explain how you know that the sequence 400, 200, 100, 50 is
geometric.
The sequence has a common ratio of 12 or 0.5 between terms.
30. Open-Ended Write a geometric sequence of at least seven terms.
any seven-term sequence with a common ratio
31. Error Analysis A student says that the geometric sequence 30, __, 120 can be
completed with 90. Is she correct? Explain.
No; the sequence can be completed with 60 with a common ratio of 2.
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Name
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Practice (continued)
9-3
Form G
Geometric Sequences
Identify each sequence as arithmetic, geometric, or neither. Then find the
next two terms.
1
32. 9, 3, 1, 3 , . . .
geometric;
33. 1, 0, 22, 25, . . .
1 1
9 , 27
neither; 29, 214
35. 23, 2, 7, 12, . . .
36. 1, 22, 25, 28, . . .
arithmetic; 17, 22
arithmetic; 211, 214
34. 2, 22, 2, 22, . . .
geometric; 2, 22
37. 1, 22, 3, 24, . . .
neither; 5, 26
Write an explicit formula for each sequence. Then generate the first
five terms.
38. a1 5 3, r 5 22
an 5 3(22)n21; 3, 26, 12,
224, 48
41. a1 5 22, r 5 23
an 5 22(23)n21; 22, 6,
218, 54, 2162
44. a1 5 9, r 5 2
an 5 9(2)n21; 9, 18, 36,
72, 144
39. a1 5 5, r 5 3
an 5 5(3)n21; 5, 15, 45,
135, 405
40. a1 5 21, r 5 4
an 5 21(4)n21; 21, 24,
216, 264, 2256
1
42. a1 5 32, r 5 20.5
43. a1 5 2187, r 5 3
n21
1
an 5 32(20.5)
; 32, 216,
an 5 2187 Q 3 R n21 ; 2187,
8, 24, 2
729, 243, 81, 27
45. a1 5 24, r 5 4
46. a1 5 0.1, r 5 22
an 5 24(4)n21; 24, 216,
an 5 0.1(22)n21; 0.1,
264, 2256, 21024
20.2, 0.4, 20.8, 1.6
47. The deer population in an area is increasing. This year, the population was
1.025 times last year’s population of 2537.
a. Assuming that the population increases at the same rate for the next few
years, write an explicit formula for the sequence. an 5 2537(1.025)n21
b. Find the expected deer population for the fourth year of the sequence. about 2732
48. You enlarge the dimensions of a picture to 150% several times. After the first
increase, the picture is 1 in. wide.
a. Write an explicit formula to model the width after each increase. an 5 1(1.5)n21
b. How wide is the photo after the 2nd increase? 1.5 in.
c. How wide is the photo after the 3rd increase? 2.25 in.
d. How wide is the photo after the 12th increase? about 86.5 in.
Find the missing terms of each geometric sequence. (Hint: The geometric mean
of positive first and fifth terms is the third term. Some terms might be negative.)
49. 12, j , j , j , 0.75
6, 3, 1.5 or 26, 3, 21.5
50. 29, j , j , j , 22304
236, 2144, 2576 or 36, 2144, 576
For the geometric sequence 6, 18, 54, 162, . . . , find the indicated term.
51. 6th term 1458
52. 19th term 2,324,522,934
53. nth term 6(3)n21
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Practice
Form K
Geometric Sequences
Determine whether each sequence is geometric. If so, find the common ratio.
1. 1, 3, 9, 27, c
2. 2, 5, 8, 11, 14, c
not geometric
Find the ratios between consecutive terms.
3
9
27
1535 9
The sequence is geometric.
3 z.
The common ratio is z 3. 22, 24, 28, 216, c
geometric; 2
4. 500, 50, 5, 0.5, c
1
geometric; 10
5. 0, 25, 50, 75, 100, c
not geometric
1
6. Open-Ended Write a geometric sequence with a common ratio of 4. Explain
how you developed the sequence.
Answers may vary. Sample: 64, 16, 4, 1, . . . . I divided the first term by 4 to get the
second term. Then I divided the second and third terms by 4.
Find the ninth term of each geometric sequence.
7. 3, 12, 48, 192, c
Use the explicit formula.
8. 2, 6, 18, 54, c
13,122
9. 1875, 375, 75, 15, c
0.0048
an 5 a1 ? rn21
a9 5 3(48)
a9 5 3(65,536)
z
a9 5 196,608
z
Find the missing terms of each geometric sequence.
10. 2, ___, ___, 128, c
z
zz
z
2 , 4 , 8, c
11. 1, z
Identify the common ratio.
an 5 a1 ? rn21
a4 5 2r421
128 5 2r3
64 5 r3
45r
8 z.
The second term is z 32 z .
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25
zz
z
36 , 12 , 4, c
12. 108, Name
Class
Date
Practice (continued)
9-3
Form K
Geometric Sequences
Find the missing term of each geometric sequence. It could be the geometric
mean or its opposite.
z
z
14. 2, w12
, 72, c
13. 5, ___, 45, c
Find the geometric mean of 5 and 45.
!xy
!45 ? 5
!225
z w15
z
z
z
1
w34 , 2 1 , c
15. 4 , 4
z
z
16. 175, w35
, 7, c
z
z
17. 1.2, w7.2
, 43.2, c
18. Error Analysis On a recent math test, your classmate was asked to find the
missing term in the geometric sequence 4, ___, 256. Her answer was 130. What
error did your classmate make? What is the correct answer?
She found the arithmetic mean of 256 and 4 rather than the geometric mean; 32
19. The bacteria population in a petri dish was 14 at the beginning of an
experiment. After 30 min, the population was 28, and after an hour the
population was 56.
a. Write an explicit definition to represent this sequence. an 5 14 ? 2n21
b. If this pattern continues, what will be the bacteria population after 4 h? 3584
20. A corporation earned a profit of $420,000 in its first year of operation. Over the
next 10 years, the company’s CEO hopes to increase the profit by 8% each year.
If the CEO reaches her goal, what will be the company’s profit in its seventh
year, to the nearest dollar? $666,487
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Standardized Test Prep
Geometric Sequences
Multiple Choice
For Exercises 1−6, choose the correct letter.
1. What is the 10th term of the geometric sequence 1, 4, 16, . . .? C
40
180,224
262,144
2,883,584
2. Which sequence is a geometric sequence? H
1, 3, 5, 7, 9, . . .
2, 4, 8, 16, 32, . . .
12, 9, 6, 3, 0, . . .
22, 26, 210, 214, 218, . . .
3. Which could be the missing term of the geometric sequence 5, __, 125, . . .? A
25
50
75
100
3
4. What could be the missing term of the geometric sequence 212, __,24 , . . .? H
24
3
26.375
4
5. In the explicit formula for the 9th term of the geometric sequence
1, 6, 36, . . . what number is a? A
1
6
36
1,679,616
6. In each successive round of a backgammon tournament, the number of
players decreases by half. If the tournament starts with 32 players, which rule
could predict the number of players in the nth round? I
32 5 (0.5)n
32 5 0.5r n21
an 5 15n21
an 5 (32)(0.5)n21
Short Response
7. What is the 6th term of the geometric sequence 100, 50, . . .? Show your work
using the explicit formula.
1
[2] 3.125; an 5 ar n21 ; an 5 100 Q 1
R n21 ; a6 5 100 Q R 5 5 3.125;
2
2
correct term with work shown
[1] incorrect term OR correct answer, without work shown
[0] incorrect answers and no work shown OR no answers given
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Enrichment
Geometric Sequences
Doubling Periods in Geometric Sequences
3
Consider the geometric sequence 3, 4 12, 6 4, 10 18, . . . .
1
1. Describe how the terms of the sequence are related. Each term is 12 times the preceding term.
2. For any term of the sequence, how many terms does it take before the value of
the term has at least doubled? 2
The doubling period of a geometric sequence is the number of terms needed to
reach a term at least twice as large as a given term. What is the doubling period for
the given sequence? 2 terms
3. Write the first ten terms of the geometric sequence a1 5 3, r 5 1.1 to two decimal
places. 3, 3.3, 3.63, 3.99, 4.39, 4.83, 5.31, 5.85, 6.43, 7.07
4. What is the doubling period for a1 5 3? for a2 5 3.3? 8 terms; 8 terms
Although the doubling period does not depend on which term is given, it does
depend on the common ratio. For what value(s) of r is the doubling period of a
geometric sequence greater than 1? 1 R | r | R 2
The idea of a doubling period applies to certain everyday situations. For example,
under optimum conditions, bacteria reproduce by splitting in two. Their numbers
increase geometrically over time. Suppose at noon on a certain day, there are 1000
bacteria in a dish. At 6 p.m. on the same day, there are 8000 bacteria.
5. If a count is taken every hour, how many terms are in the geometric sequence?
What is the common ratio? What is the doubling period? 7; !2; 2 terms
6. If a count is taken every 40 min, how many terms are in the sequence? What is
3
the common ratio? What is the doubling period? 10; "2; 3 terms
7. In both cases, how many hours does it take the bacteria to double? 2
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Name
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Date
Reteaching
9-3
Geometric Sequences
• A geometric sequence has a constant ratio between consecutive terms. This
number is called the common ratio.
• A geometric sequence can be described by a recursive formula, an 5 an21 ? r ,
or as an explicit formula, an 5 a ? r n21 .
Problem
Find the 12th term of the geometric sequence 5, 15, 45, . . . .
5, 15, 45, . . .
15
45
r 5 5 5 15 5 3
an 5 5(3)n21
Find r by calculating the common ratio between consecutive terms. This is a
geometric sequence because there is a common ratio between consecutive terms.
Substitute a 5 5 and r 5 3 into the explicit formula to find a formula for
the nth term of the sequence.
a12 5 5(3)11
Substitute n 5 12 to find the 12th term of the sequence.
a12 5 885,735
Remember to first calculate 311, then multiply by 5.
Exercises
Find the indicated term of the geometric sequence.
1
1. 4, 2, 1, . . . Find a10. 128
2 4
64
4. 1, 23 , 9 , . . . Find a7. 729
10,935
15 45
2. 5, 2 , 4 , . . . Find a8. 128
2
2
3. 6, 22, 3 , . . . Find a12. 259,049
5. 100, 200, 400, . . . Find a9.
25,600
6. 8, 32, 128, . . . Find a4. 512
Write the explicit formula for each sequence. Then generate the first five terms.
1
7. a1 5 1, r 5 2
8. a1 5 2, r 5 3
9.
1 n21
1 1 1 1
n
2
1
an 5 1 Q 2 R
; 1, 2, 4, 8, 16
an 5 2(3)
; 2, 6, 18, 54, 162
1
1
10. a1 5 1, r 5 4
11. a1 5 5, r 5 10
12.
1 n21
1 1
1
1
1 n21
1 1 1
1
; 5, 2, 20, 200, 2000
an 5 1 Q 4 R
; 1, 4, 16, 64, 256 an 5 5 Q 10 R
a1 5 12, r 5 3
an 5 12(3)n 2 1; 12, 36, 108, 324, 972
a1 5 1, r 5 13
1 1
an 5 1 Q 13 R n21; 1, 13, 19, 27
, 81
13. a1 5 5, r 5 2
14. a1 5 1, r 5 3
15. a1 5 3, r 5 6
n21
n
2
1
an 5 5(2)
; 5, 10, 20, 40, 80 an 5 1(3)
; 1, 3, 9, 27, 81 an 5 3(6)n 2 1; 3, 18, 108, 648, 3888
1
16. a1 5 3, r 5 3
17. a1 5 2, r 5 2
18. a1 5 2, r 5 2
an 5 3(3)n 2 1; 3, 9, 27, 81, 243 an 5 2(2)n 2 1; 2, 4, 8, 16, 32
an 5 2 Q 12 R n21 ; 2, 1, 12, 14, 18
1
19. a1 5 1, r 5 5
20. a1 5 3, r 5 4
21.
1 n21
1 1
1
1
n
2
1
; 3, 12, 48, 192, 768
an 5 1 Q 5 R
; 1, 5, 25, 125, 625 an 5 3(4)
a1 5 5, r 5 14
5 5
5
an 5 5 Q 14 R n21 ; 5, 54, 16
, 64, 256
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Name
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Reteaching (continued)
9-3
Geometric Sequences
Problem
From 2000 to 2009, your friend’s landlord has been allowed to raise her rent by
the same percent each year. In 2000, her rent was $1000, and in 2003, her rent was
$1092.73. What was her rent in 2009?
Step 1 Identify key information in the problem.
You know that your friend’s rent was $1000 in 2000. This means a 5 1000. You also
know that her rent in 2003 was $1092.73. This means that a4 5 1092.73. Her rent is
raised by the same percent each year, which is the same as multiplying by a constant
(e.g., a 5% increase is the same as multiplying by 1.05).
Step 2 Identify missing information.
You need to find the common ratio r in order to find the rent in 2009, a10.
Step 3 Use the explicit formula to find r.
an 5 arn 2 1
1092.73 5
(1000)r4 2 1
Write the explicit formula.
1092.73 5 1000r3
Substitute a 5 1000, a4 5 1092.73, and n 5 4.
Simplify.
1.09273 5 r3
Divide each side by 1000.
1.03 5 r
Take the cube root of both sides.
Step 4 Use the value of r to find the rent in 2009, a10.
an 5 arn 2 1
Write the explicit formula.
a10 5 (1000)(1.03)1021
a10 5
(1000)(1.03)9
a10 < 1304.77
Substitute a 5 1000, r 5 1.03, and n 5 10.
Simplify.
Compute. Round to the nearest hundredth.
Your friend’s rent was $1304.77 in 2009.
Exercises
22. An athlete is training for a bicycle race. She increases the amount she bikes by
the same percent each day. If she bikes 10 mi on the first day, and 12.1 mi on
the third day, how much will she bike on the fifth day? By what percent does
she increase the amount she bikes each day? 14.641 mi; 10%
23. By clipping coupons and eating more meals at home, your family plans to decrease
their monthly food budget by the same percent each month. If they budgeted $600
in January and $514.43 in April, how much will they budget in December? $341.28
24. From 2005 to 2009, a teen raised her babysitting rates by a fixed percent every
year. If she charged $8/h in 2005 and $10.04/h in 2007, how much did she
charge in 2009? What is her percent of increase each year? $12.59/h; 12%
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Name
9-4
Class
Date
Additional Vocabulary Support
Arithmetic Series
For Exercises 1−5, draw a line from each word or phrase in Column A to its
definition in Column B.
Column A
Column B
1. arithmetic series
A. a series that continues without end
2. finite series
B. the sum of the terms of a sequence
3. infinite series
C. a series whose terms form an arithmetic sequence
4. limits
D. the least and greatest values of n in a series
5. series
E. a series with a first and a last term
Identify the following series as finite or infinite.
finite
6. 2 1 6 1 18 1 54
7. 3 1 10 1 17 1 24 1 c
infinite
8. 2 1 10 1 50 1 250 1 c
infinite
9. Circle the arithmetic series in the group below.
1 1 4 1 7 1 10 1 13
4 1 14 1 24 1 34 1 44
4 1 6 1 10 1 12 1 16
2 1 4 1 8 1 16 1 32
0 1 12 1 24 1 36 1 48
1 1 6 1 36 1 216 1 1296
10. Complete the summation notation for the following sequence by filling in the
upper and lower limits.
u
3 1 11 1 19 1 27 1 35 1 c 1 115 5 a (8n 2 5) lower limit: n 5 1; upper limit: 15
u
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Think About a Plan
Arithmetic Series
Architecture In a 20-row theater, the number of seats in a row increases by three
with each successive row. The first row has 18 seats.
a. Write an arithmetic series to represent the number of seats in the theater.
b. Find the total seating capacity of the theater.
c. Front-row tickets for a concert cost $60. After every 5 rows, the ticket price goes down
by $5. What is the total amount of money generated by a full house?
1. Write the explicit formula for an arithmetic sequence. an 5 a1 1 (n 2 1)d
2. What are a1 and d for the sequence that represents the number of seats
in each row?
a1 5 z 18 z
d 5 z 3 z
3. Write an explicit formula for the arithmetic sequence that represents the
number of seats in each row.
an 5 3n 1 15
4. Write an arithmetic series to represent the number of seats in the theater.
20
a (3n 1 15)
n51
5. How can you use a graphing calculator to evaluate the series?
Answers may vary. Sample: Use the sum command and the sequence command
sum(seq(3N115,N,1,20))
.
6. Find the total seating capacity of the theater. 930
7. Write a series for the number of seats in each set of 5 rows.
5
10
20
15
a (3n 1 15); a (3n 1 15); a (3n 1 15); a (3n 1 15)
n51
n56
n511
n516
8. Use your graphing calculator to evaluate each series.
9. What are the ticket prices for each set of 5 rows?
120; 195; 270; 345
$60; $55; $50; $45
10. What is the total amount of money generated by a full house? $46,950
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Practice
Form G
Arithmetic Series
Find the sum of each finite arithmetic series.
1. 1 1 3 1 5 1 7 1 9 25
2. 5 1 8 1 11 1 c 1 26 124
3. 4 1 9 1 14 1 c 1 44 216
4. (210) 1 (225) 1 (240) 1 c 1 (285) 2285
5. 17 1 25 1 33 1 c 1 65 287
6. 125 1 126 1 127 1 c 1 131 896
7. A bookshelf has 7 shelves of different widths. Each shelf is narrower than the
shelf below it. The bottom three shelves are 36 in., 31 in., and 26 in. wide.
a. The shelf widths decrease by the same amount from bottom to top. What is
the width of the top shelf? 6 in.
b. What is the total shelf space of all seven shelves? 147 in.
Write each arithmetic series in summation notation.
4
n51
10. 1 1 3 1 5 1 c 1 13
6
9. 10 1 7 1 4 1 c 1 (25)
8. 4 1 8 1 12 1 16 a 4n
a (23n 1 13)
n51
7
11. 3 1 7 1 11 1 c 1 31
a (2n 2 1)
n51
8
a (4n 2 1)
n51
12. (220) 1 (225) 1 (230) 1 c 1 (265)
13. 15 1 25 1 35 1 c 1 75
7
10
a (10n 1 5)
a (25n 2 15)
n51
n51
Find the sum of each finite series.
4
6
14. a (n 2 1) 6
8
15. a (2n 2 1) 35
n51
n52
5
4
17. a (5n 1 3) 82
n51
10
4
20. a n 45
19. a (3 2 n) 23
n51
6
21. a (2n 2 3) 222
n55
n53
6
18. a (2n 1 0.5) 22
n52
16. a (n 1 25) 183
n51
22. a (3n 1 2) 62
n53
Use a graphing calculator to find the sum of each series.
15
12
23. a (n 1 3) 165
n51
25
n51
26. a (n3 1 2n) 106,275
n51
20
24. a (2n 2 1) 144
50
27. a (n2 2 4n) 37,825
n51
25. a 2n2 5740
n51
25
28. a (5n3 1 3n) 528,570
n55
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Practice (continued)
Form G
Arithmetic Series
Determine whether each list is a sequence or a series and finite or infinite.
29. 7, 12, 17, 22, 27
sequence; finite
30. 3 1 5 1 7 1 9 1 c
series; infinite
31. 8, 8.2, 8.4, 8.6, 8.8, 9.0, c
sequence; infinite
32. 1 1 5 1 9 1 c 1 21
series; finite
33. 40, 20, 10, 5, 2.5, 1.25, c
sequence; infinite
34. 10 1 20 1 30 1 40 1 50
series; finite
35. An embroidery pattern calls for five stitches in the first row and for three more
stitches in each successive row. The 25th row, which is the last row, has
77 stitches. Find the total number of stitches in the pattern. 1025 stitches
36. A marching band formation consists of 6 rows. The first row has 9 musicians,
the second has 11, the third has 13 and so on. How many musicians are in the
last row and how many musicians are there in all? 19 musicians; 84 musicians
37. Writing Explain how you can identify the difference between a series and a
sequence.
A series is the sum of terms in a sequence, which is indicated by summation
notation or addition signs.
38. a. Open-Ended Write three explicit formulas for arithmetic sequences.
b. Write the first seven terms of each related series.
c. Use summation notation to rewrite the series.
d. Evaluate each series.
Check students’ work. Sequences should be arithmetic and contain
seven terms.
39. Error Analysis A student identifies the series 10 1 15 1 20 1 25 1 30 as an
infinite arithmetic series. Is he correct? Explain.
No; the series is a finite arithmetic series. An infinite arithmetic series would
continue indefinitely.
3
40. Mental Math Use mental math to evaluate a (2n 1 1). 15
1
41. To train new employees, an employer offers a bonus after 30 work days as
follows. An employee must turn in one report on the first day; the number of
reports for each subsequent day must increase by two. What is the minimum
number of reports an employee will have to turn in over the 30 days to earn the
bonus? 900
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Class
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Practice
Form K
Arithmetic Series
Identify each list as a series or a sequence and finite or infinite.
1. 2, 6, 10, 14, c
infinite sequence
2. 1 1 4 1 7 1 10 1 13
finite series
3. 4, 10, 16, 22, 28
finite sequence
4. 5 1 12 1 19 1 26 1 33
finite series
5. 1.4 1 1.1 1 0.8 1 0.5 1 . . . 6. 22 2 11 2 20 2 29 2 c
infinite series
infinite series
Find the sum of each finite arithmetic series.
7. 1 1 3 1 5 1 c 1 99
8. 3 1 7 1 11 1 15 1 c 1 55
Find the number of terms. Find the sum.
an 5 a1 1 (n 2 1)d
99 5 1 1 (n 2 1)2
Find the number of terms. Find the sum.
n
Sn 5 2 (a1 1 an) an 5 a1 1 (n 2 1)d
50
S50 5 2 (1 1 99) 14 terms
99 5 1 1 2n 2 2
100 5 2n
5 25(100)
z
5 2500
n
Sn 5 2 (a1 1 an)
406
z
50 5 n
10. 2 1 10 1 18 1 c 1 378
9120
9. 106 1 101 1 96 1 c 1 1
1177
11. (24) 1 (29) 1 (214) 1 c 1 (299)
21030
12. Reasoning Is it possible to find the sum of an infinite arithmetic series? Explain.
No; an infinite arithmetic series has a never-ending number of terms, so it is impossible
to add them all.
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Practice (continued)
Form K
Arithmetic Series
Write each arithmetic series in summation notation.
13. 3 1 8 1 13 1 c 1 268
Find an explicit formula
for the nth term.
Find the value of n for 268.
268 5 5n 2 2
an 5 a1 1 (n 2 1)d
an 5 3 1 (n 2 1)5
270 5 5n
Write the summation notation.
u
54
z
z
22 R
a Q 5n
54 5 n
n51
an 5 5n 2 2
14. 1 1 7 1 13 1 c 1 343
15. 5 1 7 1 9 1 c 1 131
58
64
a (6n 2 5)
a (2n 1 3)
n51
n51
16. Tabitha used tiles to make the design shown at the right. The first
column has 2 tiles, the second column has 4 tiles, and the pattern
continues.
a. Write an explicit formula for the sequence. an 5 2n
b. Write the summation notation for a related series with 24 tiles 12
a (2n)
in the 12th column.
n51
c. How many tiles are in the design if there are a total of 12 columns?
156 tiles
17. Your brother is preparing for basketball season. He shot 26 baskets on the first
day that he practiced. He shot 32 baskets on the second day and 38 baskets the
day after that.
a. If this pattern continues, how many baskets will he shoot on the 30th day? 200 baskets
b. How many baskets will he have shot during those 30 days? 3390 baskets
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Standardized Test Prep
Arithmetic Series
Multiple Choice
For Exercises 1–6, choose the correct letter.
1. What is the sum of the odd integers 1 to 99? B
2450
2500
2550
4950
2. Which of the following is an infinite series? H
3, 8, 13, 18, 23
3 1 8 1 13 1 18 1 23 1 c
3 1 8 1 13 1 18 1 23
3, 8, 13, 18, 23, c
3. The high school choir is participating in a fundraising sales contest. The choir
will receive a bonus if they make 20 sales in their first week and improve their
sales by 3 in every subsequent week. What is the minimum number of sales
the choir could make in the first 12 weeks to qualify for the bonus? C
13
53
438
5015
4. What is summation notation for the series 5 1 7 1 9 1 c 1 105? F
51
51
50
a (2n 1 3)
a (n 1 3)
a (2n 1 3)
n51
n51
n51
51
a (n 1 3)
n57
100
5. What is the upper limit of the summation a (n 2 2)? D
n51
1
2
98
100
990
1980
30
6. What is the sum of the series a (2n 1 2)? H
n51
62
66
Short Response
7. What is the sum of the finite arithmetic series 2 1 4 1 6 1 c 1 50?
Show your work.
[2] 650; Sn 5 n2 (a1 1 an); S25 5 25
2 (2 1 50) 5 (12.5)(52) 5 650
[1] incorrect sum OR correct sum, without work shown
[0] incorrect answer and no work shown OR no answer given
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Enrichment
Arithmetic Series
The Gauss Trick
If your teacher asked you to add the numbers from 1 to 100, you would probably begin by
adding 1 1 2 1 3 1 c1 100, term by term from left to right. Karl Friedrich Gauss (1777–
1855) found another way. Let S represent the finite series whose sum you are trying to find.
Since addition is commutative, both equations below represent this series.
S 5 001 1 92 1 93 1 c1 98 1 99 1 100
S 5 100 1 99 1 98 1 c1 93 1 92 1 001
1. What is the sum of the left side of the first equation and the left side of the second equation?
2S
2. What is the sum of each vertically-aligned pair of quantities on the right side of
the equal signs? 101
3. How may such pairs are there? 100
4. Because each pair has the same sum, use multiplication to express the sum of
all the pairs on the right side. 100 3 101 5 10,100
5. Write an equation that states that the sum of the left sides must equal the sum of
the right sides. Solve your equation for S. 2S 5 10,100; S 5 5050
Use the technique outlined above to derive the formula for the sum of n terms of any
arithmetic series. Suppose that the series starts with the term a1 and has a common
difference of d.
6. What is the nth term, in terms of a1, d, and n? a1 1 (n 2 1)d
7. Write the sum S of the n terms of the series, where each number is written in
terms of a1 and d. Then write the sum in reverse order, lining up terms.
S 5 a1 1 (a1 1 d ) 1 c1 fa1 1 (n 2 1)d g; S 5 fa1 1 (n 2 1)d g 1 c1 (a1 1 d ) 1 a1
8. What is the sum of each vertical pair of quantities on the right side?
2a1 1 (n 2 1)d
9. How many such pairs are there? n
10. Express the sum of all the pairs using multiplication. nf2a1 1 (n 2 1)dg
11. Write an equation that states that the sum of the left sides must equal the sum of
the right sides. Solve your equation for S. 2S 5 nf2a1 1 (n 2 1)dg; S 5 nf2a1
n
1 (n 2 1)dg
2
12. Show that your equation is equivalent to S 5 2 (a1 1 an). Hint: Use your answer to Exercise 6.
nf2a1 1 (n 2 1)dg
S5
2
5
5
5
nfa1 1 a1 1 (n 2 1)dg
2
nfa1 1 fa1 1 (n 2 1)dgg
2
nfa1 1 an g
5 n2 (a1 1 an)
2
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Reteaching
Arithmetic Series
Summation notation shows the upper limit, lower limit, and explicit formula for
the terms of a series.
To find the sum of an arithmetic series written in summation notation:
n
• list the terms and add them, or use the formula Sn 5 2 (a1 1 an)
Problem
What is the sum of the series written in summation notation?
4
a. a (5 2 2n)
n52
4
a (5 2 2n)
Circle the upper and lower limits. Box the
explicit formula.
n52
n52
n53
n54
n52
n53
n54
5 2 2(2)
5 2 2(3)
5 2 2(4)
In circles, write all possible values of n, from
the lower limit to the upper limit.
Under each circle copy the explicit formula,
substituting the value in the circle for n.
4
a (5 2 2n) 5 5 2 2(2) 1 5 2 2(3) 1 5 2 2(4)
The value of the series is the sum of the
values in the boxes.
n52
1
23
5
5
(21)
1
1
(23)
Evaluate each expression.
Find the sum of the terms.
The sum of the series is 23.
15
b. a (4n 2 1)
n51
n
Use the formula Sn 5 2 (a1 1 an).
a1 5 4(1) 2 1 5 3
an 5 a15 5 4(15) 2 1 5 59
15
Sn 5 2 (3 1 59)
5 465
First, find n, a1 , and an . The upper limit is 15.
Evaluate the explicit formula at n 5 1.
Evaluate the explicit formula at n 5 15.
Substitute n 5 15, a1 5 3, and an 5 59.
Simplify.
The sum of the series is 465.
Exercises
Find the sum of each finite series.
3
4
1. a (n 2 4) 26
n51
9
3. a (3n 2 1) 93
n51
n53
5
5. a (4 2 2n) 256
n53
8
1 10
2. a 3 n 3
8
2n
4. a 3 22
n53
7
6. a 8n 120
7. a 4n 108
n51
n52
7
8. a (3 2 2n) 235
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n51
Name
Class
Date
Reteaching (continued)
9-4
Arithmetic Series
Problem
The debate club is offering a prize at the end of 10 weeks to a current member who
brings three new members for the first meeting, and then increases the number of
new members they bring each week by two thereafter. One member qualified for
the prize with the minimum number of new members. How many new members
did the member bring at Week 10? For all 10 weeks?
Step 1 Identify key information in the problem.
To win the prize, a member must bring three members to the first meeting, so a 5 3.
A member must also bring two more new members to each meeting, so d 5 2.
The contest extends for 10 weeks, so n 5 10.
Step 2 Identify the information you are trying to find.
You want to find the 10th term, a10 , and the sum of the first 10 terms, S10 .
Step 3 Use the explicit formula to find a10 .
an 5 a 1 (n 2 1)d
a10 5 3 1 (10 2 1)2
a10 5 21
Write the explicit formula.
Substitute a 5 3, d 5 2, and n 5 10.
Simplify.
To win the prize, a member brought 21 new members to a meeting at Week 10.
Step 4
Use the value of a10 to find the total number of new members brought by the winner.
Sn 5 n2 (a1 1 an)
10
S10 5 2 (3 1 21)
S10 5 120
Write the formula for the sum of an arithmetic series.
Substitute a1 5 3, a10 5 21, and n 5 10.
Simplify.
The debate club had 120 new members brought in by the winner of the contest.
Exercises
9. The seating arrangement for a recital uses 20 seats in the first row and two
additional seats in each row thereafter. How many seats will be in the eighth
row? In the ninth row? How many seats total are there in the first nine rows?
34 seats; 36 seats; 252 seats
10. With the help of a tutor, a student’s weekly quiz scores have increased during
the first four quizzes: 65, 70, 75, and 80. If the scores continue to increase at
this rate, what will be the score in the 7th week? In the 8th week? What is the
total of the first eight scores? 95; 100; 660
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Additional Vocabulary Support
Geometric Series
Problem
What is the sum of the geometric series 2 1 6 1 18 1 54 1 c1 1458?
6
18
54
2 5 6 5 18 5 3
nth term 5 1458
an 5 a1rn21
1458 5 2 ? 3n21
729 5 3n21
729 is 36 , so n 2 1 5 6 and n 5 7
Identify the common ratio and the nth term.
Use the explicit formula.
Substitute 2 for a1, 3 for r, and 1458 for an.
Divide each side by 2.
Use a calculator.
Sn 5
a1(1 2 rn)
12r
Use the sum formula.
S7 5
2(1 2 37)
123
Substitute 2 for a1, 3 for r, and 7 for n.
S7 5 2186
Exercise
What is the sum of the geometric series 1 1 4 1 16 1 64 1 c1 1024?
16
64
4
1 5 4 5 16 5 4
Identify the common ratio and the nth term .
nth term 5 1024
an 5 a1rn21
Use the explicit formula
.
1024 5 1 ? 4n21
Substitute 1 for a1, 4 for r, and 1024 for an
.
1024 5 4n21
Divide each side by 1
.
Use a calculator
.
1024 is 45, so n 2 1 5 5 and n 5 6
Sn 5
a1(1 2 rn)
12r
Use the sum formula
.
S6 5
1(1 2 46)
124
Substitute 1 for a1, 4 for r, and 6 for n
.
S6 5 1365
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41
Name
9-5
Class
Date
Think About a Plan
Geometric Series
Communications Many companies use a telephone chain to notify employees
of a closing due to bad weather. Suppose a company’s CEO calls three people.
Then each of these people calls three others, and so on.
a. Make a diagram to show the first three stages in the telephone chain. How
many calls are made at each stage?
b. Write the series that represents the total number of calls made through the
first six stages.
c. How many employees have been notified after stage six?
1. What type of diagram can you make to represent the telephone chain?
tree diagram
2. Make a diagram to show the first three stages in the telephone chain.
3. What expression represents the number of calls made at stage n? 3n
4. Write the series that represents the total number of calls made through the
first six stages. 3 1 9 1 27 1 81 1 243 1 729
5. What is the sum of this series? 1092
n
6. Write the sum formula. S 5 a1(1 2 r )
n
12r
7. Use the sum formula to find how many employees have been notified
n
after stage six. S 5 a1(1 2 r ) 5 3(1 2 36) 5 1092
n
12r
123
8. Does your answer agree with your sum from Exercise 5? yes
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42
Name
Class
Date
Practice
9-5
Form G
Geometric Series
Evaluate each finite series for the specified number of terms.
1. 40 1 20 1 10 1 c; n 5 10 79.921875
2. 4 1 12 1 36 1 c; n 5 15 28,697,812
3. 15 1 12 1 9.6 1 c; n 5 40 about 74.99
4. 27 1 9 1 3 1 c; n 5 100 about 40.5
5. 0.2 1 0.02 1 0.002 1 c; n 5 8 0.22222222 6. 100 1 200 1 400 1 c; n 5 6 6300
7. This month, your friend deposits $400 to save for a vacation. She plans to
deposit 10% more each successive month for the next 11 months. How much
will she have saved after the 12 deposits? $8553.71
Determine whether each infinite geometric series diverges or converges. State
whether each series has a sum.
3
3
9. 4 1 2 1 1 1 c
8. 3 1 2 1 4 1 c
converges; yes
converges; yes
11. 6 1 11.4 1 21.66 1 c
diverges; no
12. 220 2 8 2 3.2 2 c
converges; yes
10. 17 1 15.3 1 13.77 1 c
converges; yes
13. 50 1 70 1 98 1 c
diverges; no
Evaluate each infinite geometric series.
14. 8 1 4 1 2 1 1 1 c 16
1
15. 1 1 13 1 19 1 27
1 . . . 1.5
16. 120 1 96 1 76.8 1 61.44 1 c 600
17. 1000 1 750 1 562.5 1 421.875 1 c 4000
18. Suppose your business made a profit of $5500 the first year. If the profit
increased 20% per year, find the total profit over the first 5 yr. $40,928.80
19. The end of a pendulum travels 50 cm on its first swing. Each swing after the
first, it travels 99% as far as the preceding swing. How far will the pendulum
travel before it stops? 5000 cm
20. A seashell has chambers that are each 0.82 times the length of the enclosing
chamber. The outer chamber is 32 mm around. Find the total length of the
shell’s spiraled chambers. about 177.78 mm
21. The first year a toy manufacturer introduces a new toy, its sales total $495,000.
The company expects its sales to drop 10% each succeeding year. Find the
total expected sales in the first 6 years. Find the total expected sales if the
company offers the toy for sale for as long as anyone buys it. $2,319,367.05; $4,950,000
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43
Name
Class
9-5
Date
Practice (continued)
Form G
Geometric Series
Determine whether each series is arithmetic or geometric. Then evaluate the
series for the specified number of terms.
1
1
1
1
23. 8 1 16 1 32 1 64 1 c; n 5 8
22. 2 1 5 1 8 1 11 1 c; n 5 9
arithmetic; 126
255
geometric; 1024
24. 23 1 6 2 12 1 24 2 c; n 5 10
geometric; 1023
25. 22 1 2 1 6 1 10 1 c; n 5 12
arithmetic; 240
26. 4 1 8 1 16 1 32 1 c; n 5 15
geometric; 131,068
27. 5 1 10 1 15 1 20 1 c; n 5 20
arithmetic; 1050
Evaluate each infinite series that has a sum.
`
2
28. a 5 Q 3 R n21
`
`
29. a (22. 1)n21
1
30. a Q 22 R n21
`
5
31. a 2 Q 3 R n21
n51
n51
n51
n51
15
no sum
2
3
no sum
32. Open Ended Write an infinite geometric series that converges to 2.
Show your work. Check students’ work.
Find the specified value for each infinite geometric series.
25
33. a1 5 5, S 5 3 , find r 2
5
1
34. S 5 108, r 5 3 , find a1 72
35. a1 5 3, S 5 12, find r 0.75
36. S 5 840, r 5 0.5, find a1 420
37. Error Analysis Your friend says that an infinite geometric series cannot have
a sum because it’s infinite. You say that it is possible for an infinite geometric
series to have a sum. Who is correct? Explain.
You are; an infinite geometric series with »r… less than 1 has a series of
partial sums that converges towards a number.
38. Writing Describe in general terms how you would find the sum of a finite
geometric series. Identify the first term, common ratio, and nth term.
Use the explicit formula to find n. Then, use the sum formula with the
first term, common ratio, and n to find the sum of the series.
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Name
Class
9-5
Date
Practice
Form K
Geometric Series
Find the sum of each finite geometric series.
1. 2 1 6 1 18 1 c 1 4374
Find the number of terms. Use the sum formula.
n
Sn 5 a1(1 2 r )
an 5 a1rn21
4374 5 2 ? 3n21
12r
2(1 2 38)
S8 5 1 2 3
z
2187 5 3n21
37
2. 1 1 2 1 4 1 c 1 2048
4095
5 6560
z
5 2187
n58
1
3. 8 1 4 1 2 1 c 1 256
N 16
4. 3 1 9 1 27 1 . . . 1 6561
9840
5. 24 2 8 2 16 2 c 2 2048
24092
6. Find the sum of the geometric series 2 2 4 1 8 2 16 1 c 1 8192. Explain
how you found the sum.
5462; I used the explicit formula to determine that there are 13 terms in the
series. Then I used the sum formula to determine the sum.
7. A family farm produced 2400 ears of corn in its first year. For each of the next
9 yr, the farm increased its yearly corn production by 15%. How many ears of
corn did the farm produce during this 10-yr period? 48,729
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Name
9-5
Class
Date
Practice (continued)
Form K
Geometric Series
Determine whether each infinite geometric series diverges or converges. Find
the sum if the series converges.
1
1
8. 1 1 4 1 16 1 c
Because ur u 5 P 14 P , 1, the series converges.
a
1
1
113
S 5 1 21 r 5
1 5 3 5 124
4
z
1
1
1
10. 2 1 16 1 128 1 c
converges; 47
9. 2 1 8 1 32 1 c
diverges
z
3
9
1
11. 4 1 8 1 16 1 c
diverges
2
2
12. 2 2 5 1 25 2 c
converges; 123
13. Your classmate is trying to cut down on the amount of time he spends watching
television. In January, he spent a total of 3600 min watching television.
He watched television for 3240 min in February and 2916 min in March. If this
pattern continues, how many minutes of television will he watch this year?
about 25,833 min
14. Your math teacher asks you to choose between two offers. The first offer is to
receive one penny on the first day, 3 pennies on the second day, 9 pennies on
the third day, and so on, for 14 days. The second offer is to receive 4 pennies
on the first day, 8 pennies on the second day, 16 pennies on the third day, and
so on, for 14 days. Which offer is better? What is the difference between the
total amounts received? the first offer; 2,325,952 pennies
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Name
Class
Date
Standardized Test Prep
9-5
Geometric Series
Gridded Response
Solve each exercise and enter your answer in the grid provided.
20
1 n
1. What is the value of a1 in the series a 3 Q 2 R ?
n50
2. What is the sum of the geometric series 2 1 6 1 18 1 c1 486?
3. A community organizes a phone tree in order to alert each family of
emergencies. In the first stage, one person calls five families. In the second
stage, each of the five families calls another five families, and so on. How
many stages need to be reached before 600 families or more are called?
4. What is the approximate whole number sum for the finite geometric series
5
n
1
a 8Q4 R ?
n50
1
1
5. What is the sum of the geometric series 1 1 3 1 9 1 c? Enter your answer
as a fraction.
Answers
1.
0
–
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
2.
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
1 1
–
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
3.
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
41
–
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
4.
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
0
–
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
5.
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
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6
–
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
Name
9-5
Class
Date
Enrichment
Geometric Series
An infinite geometric series converges if the absolute value of the common ratio is
less than 1 ( u r u , 1). A power series is an infinite series where each term depends
on a variable x. Each value of x will give you a specific infinite series, which may
converge or diverge.
1
1
1. Evaluate the expression 1 2 x for x 5 2 and for x 5 2 . 21; 2
2. You can evaluate many expressions with a power series called a Taylor series.
1
To evaluate 1 2 x , you use the Taylor series 1 1 x 1`x2 1 x3 1 c. What is
n
this infinite series written in summation notation? a x
n50
3. Determine the sum of the first five terms of the Taylor series
1
1 1 x 1 x2 1 x3 1 cfor x 5 2 and for x 5 2 . 31; 1.9375
4. For which value of x is your computation above a better approximation for the
1
value of the expression 12x ? What might need to be true about the value of x
in order for this Taylor series to converge to the value of this expression? 12;»x… R 1
5. You can use a different Taylor series to evaluate ex . Write the Taylor series
` n
x
x
x2
x3
1 1 1! 1 2! 1 3! 1 cin summation notation. a n!
n50
1
6. Evaluate ex for x 5 2 . Evaluate the first four terms of the Taylor series
x
x2
x3
1
1 1 1! 1 2! 1 3! 1 cfor x 5 2 . Round your answers to the nearest
thousandth. 1.649; 1.646
7. Does the Taylor series for ex still converge if u x u $ 1? Does it give you the
same value as the function y 5 ex? Explain your reasoning.
Answers may vary. Sample: Yes; yes; for any value of x, the terms eventually decrease
rapidly to 0. If you add up enough terms, you will get a good approximation.
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Name
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Date
Reteaching
9-5
Geometric Series
a (1 2 rn)
• The sum of a finite geometric series is Sn 5 1 1 2 r , where a1 is the first
term, r is the common ratio, and n is the number of terms.
a
• The sum of an infinite geometric series with u r u , 1 is S 5 1 21 r , where a1 is
the first term and r is the common ratio. If u r u $ 1, then the series has no sum.
Problem
What is the sum of the first ten terms of the geometric series
8 1 16 1 32 1 64 1 128 1 c?
a1 is the first term in the series.
a1 5 8
16
32
64
128
r 5 8 5 16 5 32 5 64 5 2
n is the number of terms in the series to be added together.
n 5 10
8(1 2 210)
S10 5 1 2 2
5
Simplify the ratio formed by any two consecutive terms to find r.
Substitute a1 5 8, r 5 2, and n 5 10 into the formula for the sum
of a finite geometric series.
8(21023)
21
Simplify inside the parentheses.
Simplify.
5 8184
Exercises
Evaluate the finite series for the specified number of terms.
1. 3 1 12 1 48 1 192 1 c; n 5 6 4095
1
1
2. 8 1 2 1 2 1 8 1 c; n 5 5 341
32
635
3. 210 2 5 2 2.5 2 1.25 2 c; n 5 7 2 32
5
5
3415
4. 10 1 (25) 1 2 1 Q 24 R 1 c; n 5 11 512
Evaluate each infinite geometric series.
5. 10 1 5 1 2.5 1 c 20
7
49
2
4
1
11
6. 21 1 11 2 121 1 c 213 7. 4 1 32 1 256 1 c 2
1
1
2
5
8. 2 2 5 1 25 2 c 14
1
1
1
1
9. 26 1 12 2 24 1 c 29
4
11. 12 1 4 1 3 1 c 18
1
1
1
1
12. 4 2 8 1 16 2 c 6
64
10. 20 1 16 1 5 1 c 100
2
2
2
5
13. 3 1 15 1 75 1 c 6
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Name
Class
9-5
Date
Reteaching (continued)
Geometric Series
Problem
Your neighbor hosts a family reunion every year. In 2000, it costs $1500 to host
the reunion. Their expenses have decreased by 10% per year by asking family
members to contribute food and party supplies.
a. What is a rule for the cost of the family reunion?
b. What was the cost of the reunion in 2005?
c. What was the total cost for hosting the family reunions from 2000 to 2009?
The cost is a geometric sequence that decreases by the same percent each year.
an 5 arn21
an 5 (1500)(0.90)n21
Write the explicit formula.
Substitute a1 5 1500, r 5 1 2 0.10 5 0.90 in the explicit formula.
To find the cost of the reunion in 2005 (n 5 6), substitute values into the explicit
formula.
an 5 arn21
an 5 (1500)(0.90)621
Write the explicit formula.
Substitute a1 5 1500, r 5 0.90, n 5 6 in the formula.
an < 886
Simplify.
The cost of hosting the reunion in 2005 was $886.
10
To find the total of hosting the reunions from 2000 to 2009, a (1500)(0.90)n21 ,
n51
find the sum of the geometric series.
Sn 5
S10 5
a1(1 2 rn)
12r
1500(1 2 0.9010)
1 2 0.90
Write the formula for the sum of a geometric series.
Substitute a1 5 1500, r 5 0.90, n 5 10 in the formula.
S10 < 9770
Simplify.
The cost of hosting the reunions from 2000 to 2009 was $9770.
Exercise
14. In 1990, a vacation package cost $400. The cost has increased 10% per year.
a. What are the values of a1 and r? a1 5 400, r 5 1.1
b. What is a rule for the cost of the vacation? an 5 (400)(1.10)n21
c. What was cost of the vacation in 1995? $644.20
d. What was the total cost of the vacations from 1990 to 1999? $6374.97
e. If the pattern continued until 2009, what was the total cost of the vacations? $22,910
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Name
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Date
Chapter 9 Quiz 1
Form G
Lessons 9-1 through 9-2
Do you know HOW?
Find the first five terms of each sequence.
1. an 5 n2 1 2n 3, 8, 15, 24, 35
2. an 5 n 2 6 25, 24, 23, 22, 21
Find the seventh term of each sequence.
3. 10, 9, 7, 4, c 211
4. 2, 4, 8, 16, c 128
Determine whether each sequence is arithmetic. If so, identify the common
difference.
5. 13, 19, 25, 31, c yes; d 5 6
6. 16, 24, 36, 54, c no
Find the missing term of each arithmetic sequence.
7. 4, ___ , 24, 34, c 14
8. 100, ___, 92, c 96
Do you UNDERSTAND?
9. Vocabulary Explain what it means for a formula to be an explicit formula.
An explicit formula describes the nth term in a sequence using n.
10. Open-Ended Give an example of an arithmetic sequence.
Check students’ work.
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Name
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Date
Chapter 9 Quiz 2
Form G
Lessons 9-3 through 9-5
Do you know HOW?
Find the eighth term of each geometric sequence.
5
1. 4, 8, 16, c 512
2. 20,480; 5120; 1280; c 4
Find the seventh term of each sequence.
3. 2, 4, 8, 14, c 44
4. 1, 23, 9, 227, c 729
Determine whether each sequence is arithmetic or geometric. Then evaluate the finite
series for the specified number of terms.
5. 1 1 3 1 9 1 c; n 5 8
geometric; 3280
6. 25 1 32 1 39 1 c; n 5 12
arithmetic; 762
Evaluate each infinite geometric series.
`
`
1 n21 2
7. a Q 22 R
3
8. a 3(0.4)n21 5
n51
n51
Do you UNDERSTAND?
9. Open-Ended Write an arithmetic series that has a negative sum.
Answers may vary. Sample: 1.2 1 0.2 2 0.8 2 . . . 2 8.8 5 241.8
10. Reasoning Can an infinite geometric series converge when the common
ratio is greater than 1? Explain. Give an example.
Answers may vary. Sample: No, the terms of the series grow in absolute
value so they cannot have a finite sum; two possible series with r 5 2 are
3 1 6 1 12 1 24 1 . . . and (21) 1 (22) 1 (24) 1 . . . .
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Chapter 9 Test
Form G
Do you know HOW?
Write a recursive definition and an explicit formula for each sequence. Then find a10.
1. 41, 46, 51, 61, c
2. 1, 10, 100, 1000, 10000, c
an 5 an21 1 5 where
an 5 10an21 where
a1 5 41; an 5 41 1 5(n 2 1)
a1 5 1; an 5 1(10)n21;
or an 5 36 1 5n; 86
1,000,000,000
3. 3, 6, 12, 24, 48, c
an 5 2an21 where
a1 5 3; an 5 3(2)n21; 1536
Find the first five terms in each arithmetic sequence.
4. an 5 3n 1 2 5, 8, 11, 14, 17
5. an 5 n 1 5 6, 7, 8, 9, 10
6. an 5 12n 12, 24, 36, 48, 60
7. an 5 2n 1 10 9, 8, 7, 6, 5
Determine whether each sequence is arithmetic, geometric, or neither. Then
find the ninth term.
8. 3, 12, 48, 192, c geometric; 196,608
9. 22, 27, 212, 217, c arithmetic; 242
7
1
11. 2, 2, 2, 5, c arithmetic; 25
2
2 2
2
10. 10, 2, 5, 25, c geometric; 78,125
Find the missing term of each geometric sequence. It could be its geometric
mean or its opposite.
12. 16, 7, 4 6 8
13. 25, 7, 225 6 75
14. 2, 7, 50 6 10
15. 1, 7, 49 6 7
3
16. 4 , 7, 3 6 32
17. 36, 7, 4 6 12
Find the sum of each finite series.
5
8
19. a 3n 9840
18. a (n 2 1) 10
n51
n51
15
20
20. a (3n 1 1) 375
21. a (5 2 n) 2110
n51
n51
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Chapter 9 Test (continued)
Form G
Evaluate each infinite geometric series.
22. 30 1 22.5 1 16.875 1 c 120
23. 15 2 3 1 0.6 2 0.12 1 c 12.5
1
25. 4 2 2 1 1 2 2 1 c 83
5
5
27. 25 2 2 2 4 2 c 210
24. 12 1 6 1 3 1 c 24
1
1
26. 2132 1 9 2 6 1 c 28 10
Determine whether each series is arithmetic or geometric. Then evaluate the
finite series for the specified number of terms.
28. 5 1 9 1 13 1 17 1 c; n 5 10
arithmetic; 230
29. 35 1 70 1 140 1 280 1 c; n 5 7
geometric; 4445
30. 6 1 (218) 1 54 1 (2162) 1 c; n 5 8
geometric; 29840
31. 8 1 11 1 14 1 17 1 c; n 5 6
arithmetic; 93
32. 10 1 8 1 6 1 4 1 c; n 5 10
arithmetic; 10
4
4
33. 20 1 4 1 5 1 25 1 c; n 5 6
15,624
geometric; 625
34. On October 1, a gardener plants 20 bulbs. On October 2, she plants 23 bulbs.
On October 3, she plants 26 bulbs. She continues in this pattern until October 15,
when she plants the last bulbs.
a. Write an explicit formula to model the number of bulbs she plants each day. an 5 20 1 3(n 2 1)
b. Write a recursive definition to model the number of bulbs she plants each day. an 5 an21 1 3
where a1 5 20
c. How many bulbs will the gardener plant on October 15? 62 bulbs
d. What is the total number of bulbs she plants from October 1 to October 15, inclusive? 615 bulbs
35. Suppose you are building 10 steps with 6 concrete blocks in the top step and 60 blocks
in the bottom step. If the number of blocks in each step forms an arithmetic sequence,
find the total number of concrete blocks needed to build the steps. 330 blocks
Do you UNDERSTAND?
36. Writing Explain why an infinite geometric series with r 5 1 diverges. Include
an example in your explanation.
`
Answers may vary. Sample: a 5(1)n21 5 5 1 5 1 5 1 . . .;The series diverges
n51
because the number 5 is added an infinite number of times
37. Open-Ended Write a sequence and describe it using both an explicit definition
and a recursive formula.
Answers may vary. Sample: 30; 300; 3000; 30,000; 300,000; . . .; an 5 10an21
where a1 5 30; an 5 30(10)n21
38. Reasoning What does a recursive definition have that an explicit formula does not? Explain.
Answers may vary. Sample: A recursive definition contains an initial condition as well
as a formula for how to move from one term to the other. An explicit formula describes
the nth term in terms of n.
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Name
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Date
Chapter 9 Quiz 1
Form K
Lessons 9–1 through 9–2
Do you know HOW?
Find the first four terms of each sequence.
2. an 5 n3 1 5
6, 13, 32, 69
1. an 5 5n 2 2
3, 8, 13, 18
1
3. an 5 22 n
2 12, 21, 232, 22
Write a recursive definition for each sequence.
4. 80, 40, 20, 10, c
a1 5 80; an 5
1
2 an21
5. 4, 10, 16, 22, c
a1 5 4; an 5 an21 1 6
6. 3, 21, 147, 1029, c
a1 5 3; an 5 7an21
Find the 18th term of each arithmetic sequence.
7. 5, 9, 13, 17, c
73
8. 4, 1, 22, 25, c
247
9. 1.2, 1.6, 2, 2.4, c
8
Use the arithmetic mean to find the missing term in each arithmetic sequence.
z
z
10. c6, 17
, 28, c
z
z
11. c2, 26
, 214, c
z
z
12. c1.4, 4.1
, 6.8, c
Do you UNDERSTAND?
13. Writing Describe the difference between a recursive definition and an
explicit definition of a sequence. A recursive definition relates each term to the next.
An explicit definition describes the nth term of a sequence using the number n.
14. Tim takes the stairs up to his office. He enters the ground floor of the building
and climbs 12 steps to reach the first floor. He climbs a total of 24 steps to
reach the second floor and 36 steps to reach the third floor. How many steps
will Tim climb to reach his office on the 16th floor? 192 steps
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Name
Class
Date
Chapter 9 Quiz 2
Form K
Lessons 9–3 through 9–5
Do you know HOW?
Find the eighth term of each geometric sequence.
1. 4, 12, 36, 108, c
8748
1 1
2. 2, 1, 2 , 4 , c
3. 0.04, 0.2, 1, 5, c
3125
1
64
Find the sum of each finite arithmetic series.
5. 6 1 12 1 18 1 c 1 108
1026
4. 3 1 6 1 9 1 c 1 72
900
6. (22) 1 (27) 1 (212) 1 c 1 (2102)
21092
Write each arithmetic series in summation notation.
7. 1 1 5 1 9 1 c 1 85
8. 5 1 11 1 17 1 c 1 371
22
62
a (4n 2 3)
a (6n 2 1)
n51
n51
9. 212 1 204 1 196 1 c 1 (220)
30
a (220 2 8n)
n51
Find the sum of each finite geometric series.
1
1
1
1
11. 6 1 12 1 24 1 c 1 384
about 0.33
10. 5 1 15 1 45 1 c 1 10,935
16,400
12. 1 2 4 1 16 2 c 2 16,384
213,107
Do you UNDERSTAND?
13. A guitar-making company produced 60 guitars this month. The company
plans to increase production by 8% each month for the next 9 months. How
many guitars will they produce during this 10-month period? about 869 guitars
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Date
Chapter 9 Test
Form K
Do you know HOW?
Find the first five terms of each sequence.
2. an 5 n2 1 n
2, 6, 12, 20, 30
1. an 5 3n 1 4
7, 10, 13, 16, 19
1
3. an 5 2 n 2 2
−1.5, −1, −0.5, 0, 0.5
Write an explicit formula for each sequence. Then find the 12th term.
4. 2, 6, 12, 20, c
an 5 n(n 1 1); 156
5. 2.5, 3, 3.5, 4, c
an 5 12 n 1 2; 8
6. 2, 5, 10, 17, 26, c
an 5 n2 1 1; 145
Find the 20th term of each arithmetic sequence.
7. 2, 5, 8, 11, c
59
8. 56, 50, 44, 38, c
−58
9. 2.2, 2.6, 3, 3.4, c
9.8
Do you UNDERSTAND?
10. Writing Find the missing term in the arithmetic sequence below.
Then explain how you found the term.
c12, z 28
z , 44, c
Answers may vary. Sample: First, I found the sum of 12 and 44, which is 56. Then I
divided the sum by 2 to find the missing term, 28.
11. Reasoning Rita must find the 35th term in the sequence that begins
2, 9, 16, 23, c. She needs to find the answer as fast as possible. Should
Rita use a recursive definition or an explicit formula? Why? explicit formula; using a
recursive definition will require her to go through many iterations of the
definition. If she uses an explicit formula, she will be able to substitute the value
into the formula to find the answer.
12. A bus has 6 people on it as it pulls out of the station to begin its route. After
one stop, there are 11 people on the bus. After the second stop, there are 16
people on the bus. If this pattern continues, how many people will be on the
bus after 10 stops? 51 people
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Date
Chapter 9 Test (continued)
Form K
Do you know HOW?
Find the 9th term of each geometric sequence.
13. 5, 10, 20, 40, c
1280
14. 32, 28, 2, 20.5, . . .
N 0.0005
15. 22, 210, 250, 2250, c
−781,250
Find the sum of each finite arithmetic series.
16. 8 1 12 1 16 1 c 1 116
1736
17. (23) 1 (29) 1 (215) 1 c 1 (2201)
−3468
18. 1 1 5 1 9 1 c 1 157
3160
Determine whether each infinite geometric series diverges or converges. If the
series converges, state the sum.
1
19. 4, 2, 1, 2 , . . .
converges; 8
20. 6, 18, 54, 162
diverges
21. 5, 21, 0.2, 20.04, . . .
converges; 4.16
Do you UNDERSTAND?
22. Error Analysis Your friend calculated the sum of the finite geometric series
2 1 8 1 32 1 c 1 32,768. Her answer was 131,080. What error did she make?
What is the correct sum?
She used the formula for the sum of an arithmetic series rather than the sum of a
geometric series; 43,690
23. Writing Find the possible values of the missing term in the following
geometric sequence, and explain how you found the answer.
6, z ±24
z , 96, c
Answers may vary. Sample: First, I found the product of 96 and 6, which is 576.
Then I found the square root of 576, which is ±24.
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Date
Chapter 9 Performance Tasks
Task 1
a. Use your graphing calculator to graph the function f(x) 5 2x over the domain
5x | x $ 06.
b. Use the TABLE feature on your calculator to make a table of values of the
function f for the set of x-values 1, 2, 3, . . . .
c. Determine whether the sequence of function values is arithmetic, geometric,
or neither. Justify your response. geometric; There is a common ratio of 2.
d. Write a recursive definition and an explicit formula for the sequence of function
values. an 5 2an21 where a1 5 2; an 5 2n
e. Find three terms of the sequence between 512 and 8192, and identify these
as arithmetic or geometric means. Explain your reasoning.
Y1 5 2ˆX
X55
X
1
2
3
4
5
6
X50
Y 5 32
Y1
2
4
8
16
32
64
1024, 2048, and 4096; geometric means; 2048 is the square root of the
product of 512 and 8192, 1024 is the square root of the product of 512 and
2048, and 4096 is the square root of the product of 2048 and 8192.
[4] Student correctly uses calculator to view the function and constructs table
of values using positive integers for x-values. Student correctly identifies the
sequence as geometric and justifies answer. Student correctly writes recursive definition
and explicit formula for the sequence, finds three terms, and identifies these as
geometric means with justification.
[3] Student correctly completes parts (a), (b), and (c). Justification may not be fully developed.
Student completes parts (d) and (e) with only minor errors and some justification.
[2] Student correctly completes parts (a), (b), and (c). Justification is not given. Student
writes recursive definition and explicit formula with one or more errors. Student finds
three terms with one or more major errors. Justification is not given.
[1] Student determines minimal and/or incorrect information about the sequence, its
formulas, and the geometric means. There are major errors in logic.
[0] Student makes no attempt, or no response is given.
Task 2
a. Determine whether the sequence 27, 9, 3, 1, . . . is geometric, arithmetic, or neither.
Justify your response. geometric; There is a common ratio of 13 .
an 5 13 an21 where
b. Write a recursive definition and an explicit formula for this sequence.
a1 5 27; an 5 27 Q 13 R n21
10
c. Use summation notation to write the series related to the first ten terms
1 n21 29,524
; 729
a 27 Q 3 R
of the sequence give in part (a). Then evaluate this series.
n51
d. Use summation notation to write the series related to the infinite
`
1 n21
; converges; 81
sequence given in part (a). Determine whether this series diverges or a 27 Q 3 R
2
n51
coverages. If the series converages, find its sum.
e. Describe a real-world situation that can be modeled by this sequence. Answers may vary.
[4] Student correctly determines that the sequence is geometric. Student correctly finds
a recursive definition and explicit formula for the sequence. Student correctly writes
the series using summation notation, finds the sum of the first ten terms of the series,
determines that the infinite series converges, and correctly determines the sum. Student
describes a feasible real-world situation.
[3] Student completes all parts with only minor errors. Student describes a feasible
real-world situation.
[2] Student completes all parts with one or more major errors.
[1] Student determines minimal and/or incorrect information about the sequence and
series, their recursive definitions and explicit formulas, and the series in summation
notation. There are major errors in logic.
[0] Student makes no attempt, or no response is given.
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Name
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Date
Chapter 9 Performance Tasks (continued)
Task 3
a. Determine whether the sequence 2, 8, 14, 20, 26,cis arithmetic geometric, or neither.
Justify your response. Arithmetic; there is a common difference of 6. an 5 6 1 an21 where
b. Write a recursive definition and an explicit formula for this sequence. a1 5 2; an 5 2 1 6(n 2 1)
c. Find three terms of the sequence between 62 and 86, and identify 68, 74, and 80; arithmetic
these as arithmetic or geometric means. Explain your means; 74 is the average of 62
reasoning. and 86, 68 is the average of 62 and 74, and 80 is the average of 74 and 86.
`
d. Use summation notation to write the series related to the infinite sequence
a (6n 2 4); 290
given in part (a). Find the sum of the first ten terms of the series.
n51
e. Describe a real-world situation that can be modeled by the sequence
given in part (a). Check students’ work.
[4] Student uses a calculator to view the function and constructs a table of values using
positive integers for x-values. Student correctly identifies the sequence as arithmetic
and justifies answer. Student writes a recursive definition and an explicit formula for the
sequence, finds three terms, and identifies these as arithmetic means with justification.
Student uses summation notation to write a series and correctly finds the sum of the
first ten terms. Student describes a real-world situation.
[3] Student completes all parts with minor errors.
[2] Student makes major errors in one or more parts.
[1] Student determines minimal and/or incorrect information about the sequence, its
formulas, and the geometric means. There are major errors in logic.
[0] Student makes no attempt, or no response is given.
Task 4
a. Graph the function f (x) 5 20.5x2 1 4.5 for the domain 23 # x # 3 using
your graphing calculator.
b. Carefully draw the graph of the function on a sheet of graph
paper. Check student’s drawing.
c. Draw and use inscribed rectangles 1 unit wide to approximate the area
23
under the curve for the given interval. 13 units2
d. Use e f (x)dx feature from the CALC menu of your graphing calculator
to determine the area under the curve for the given interval. 18 units2
10
3
25
[4] Student correctly graphs the function over the designated domain on a graphing
calculator. Student draws a neat graph of the function on graph paper. Student makes
a close estimate of the area under the curve using rectangles. Student correctly uses a
graphing calculator to find the area under the curve.
[3] Student completes all parts with only minor errors.
[2] Student makes major errors in one or more parts.
[1] Student determines minimal and/or incorrect information about the graph and the area
under it. There are major errors in logic.
[0] Student makes no attempt, or no response is given.
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Date
Chapter 9 Cumulative Review
Multiple Choice
For Exercises 1−8, choose the correct letter.
1. What is the y-intercept of y 5 0.75(3)x? A
(0, 0.75)
(0, 2.25)
x2 1 8x 1 15
in simplest form? G
x2 2 x 2 12
x15
x15
x24
x13
(3, 0)
(4, 0)
x15
x14
(x 1 5)(x 1 3)
(x 2 4)(x 1 3)
3
!
x
3 2
"
x
2. What is
3
"x2
3. Which expression is equivalent to 6 ? C
"x2
2
!x
2
"x23
4. How is the polynomial 2x2 2 x3 1 4x 1 17 classified by degree? H
linear
quadratic
cubic
quartic
5. The discriminant of a quadratic equation has a value of 0. Which of the
following is true? A
There is one real solution.
There is one complex solution.
There are no real solutions.
There are two complex solutions.
6. Which of these does not have the same value as the others? G
log28
log39
log464
log5125
7. Which inequality is graphed? A
y#x14
y$x14
y#x24
y,x14
8
6
4
2
⫺6 ⫺4 ⫺2 O
8. If f (x) 5 4x 1 1 and g(x) 5 2x2 , what is the value
of g( f (28))? F
1922
513
x
2 4 6
⫺4
257
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y
2127
Name
Class
Date
Chapter 9 Cumulative Review (continued)
Short Response
9. Graph the system of inequalities e
y , 2x 2 1
.
y $ 2x 1 3
10. Describe how the graph of y 5 log 3(x 2 2) 1 5 compares to the
graph of the parent function.
8
6
4
2
⫺4 ⫺2 O
y
x
2 4 6 8
⫺4
The graph of y 5 log3(x 2 2) 1 5 is a shift of the graph of the
parent function y 5 log3x to the right two units and up five units.
11. How can the relationship between variables in the table be described?
The variables in the table, x and y, have an inverse variation
x
y
relationship. As x-values increase, y-values decrease, but the
product of their pairs remains constant.
1
20
2
10
4
5
5
4
12. Use the sequence 100, 95, 90, 85, . . . This is an arithmetic sequence in which each term is
a. Describe the sequence in words. five less than the previous one.
b. Find the next three terms. 80, 75, 70
13. Water leaks from a 10,000-gal tank at a rate of 5 gal/h. Write a linear model for the situation
and use it to find the amount of water in the tank after 24 h. w 5 25t 1 10,000; 9880 gal
Extended Response
14. You have a coupon for $10 off a CD. You also get a 20% discount if you show
your membership card in the CD club. How much more would you pay if the
cashier applies the coupon first? Use composite functions. Show your work.
[4] $2; student defines both functions and subtracts one from the other correctly.
[3] Student defines both functions and subtracts one from the other with minor errors.
[2] Student determines minimal and/or incorrect information about the functions and
does not subtract one from the other. There are major errors in logic.
[1] Student provides incorrect information. No work is shown.
[0] Student makes no attempt or no response is given.
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TEACHER INSTRUCTIONS
Chapter 9 Project Teacher Notes: Get the Picture
About the Project
The Chapter Project gives students an opportunity to use sequences, explicit
formulas, and recursive formulas to change the sizes of drawings and photos. They
investigate perspective, the use of grids to enlarge and reduce, and ways to crop,
enlarge, and reduce photographs.
Introducing the Project
• Ask students if they have ever seen artists draw buildings or other objects that
appear to recede in the distance.
• Ask them why it appears that railroad track rails get closer together when we
look at them in the distance.
• Explain that they will investigate the concepts of perspective and vanishing
points, and the mathematics involved in enlarging, reducing, and cropping
pictures and photographs.
Activity 1: Researching
Students research perspective, create drawings in perspective, write arithmetic
sequences, and determine explicit or recursive formulas for their sequences.
Activity 2: Designing
Students use grid paper to enlarge designs. They use the same ratios repeatedly
to draw lengths which form geometric sequences. They then write explicit or
recursive formulas for their sequences.
Activity 3: Analyzing
Students crop photos. Then they enlarge the cropped portions, writing sequences
for the widths of the enlargements.
Finishing the Project
You may wish to plan a project day on which students share their completed
projects. Encourage students to explain their processes as well as their results.
• Have students review their methods for writing explicit and recursive formulas
for arithmetic and geometric sequences.
• Ask groups to share their insights that resulted from completing the project,
such as any shortcuts they found for making their drawings or writing
formulas.
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Date
Chapter 9 Project: Get the Picture
Beginning the Chapter Project
When a book is being made, artists, designers, and photographers work with
writers and editors to make the pages visually attractive. These professionals often
work with patterns involving arithmetic and geometric sequences.
In this project, you will see how perspective affects perceived lengths and
distances. You will use grids to change the sizes of drawings. You also will learn
how a designer crops a photo, then enlarges or reduces it.
Activities
Activity 1: Researching
Research the concepts of one- and two-point perspective and
vanishing points in art.
• Measure the lengths of the arrows shown at the right.
What is the relationship between these lengths? How
does this relate to your research on perspective?
• Trace the four arrows at the right, moving the paper to the
left after tracing the longest arrow so that it is further away
from the others than it is now. What do you notice?
• Make a simple drawing of three or more similar objects
whose lengths can be represented by an arithmetic
sequence. Write the corresponding arithmetic sequence,
and a recursive or explicit formula for that sequence.
• Check students’ work; answers may vary. Sample: The lengths form an arithmetic sequence; answers may
vary. Sample: The lines of sight along the tops and bottoms of the arrows meet at a vanishing point.
• Check students’ work; answers may vary. Sample: There is no longer a vanishing point.
• Check students’ work.
Activity 2: Designing
Figure 1
Figure 2
When a book is made, a designer or artist may change the size
of an original sketch to fit the space available on a page. One
way to change the dimensions of a sketch is to use graph paper
with different size squares.
• Draw a figure or design on a sheet of graph paper. Label this
Figure 1 and record its approximate dimensions.
• Enlarge the original figure by copying each portion of Figure 1,
square by square, onto larger squares. Label this Figure 2 and record its dimensions.
• Use a ratio to compare the dimensions of Figure 1 to the dimensions of Figure 2. If the
same ratio is used to enlarge Figure 2, what would the dimensions of the new figure be?
Draw this figure, label it Figure 3, and record its dimensions.
• Explain why the lengths of the three figures form a geometric sequence.
• Write a geometric sequence corresponding to these lengths, and a recursive or explicit
formula for that sequence. Check students’ work.
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Chapter 9 Project: Get the Picture (continued)
Activity 3: Analyzing
Photographs are often cropped so that only part of the photograph
remains. Then, this cropped portion can be reduced or enlarged.
Choose a photograph in a textbook. Place a piece of paper over the
photograph, trace its original size, and draw a rectangle to indicate
a portion of the photograph that you would like to crop. Draw a
diagonal from the lower left corner to the upper right corner of the
rectangular cropped area. If this diagonal is extended through the
upper right corner of the cropped area, and a point selected anywhere along the
diagonal or its extension, then the rectangle having the chosen point as its upper
right corner (and the same lower left corner as the original cropped area) will have
dimensions that are proportional to the dimensions of the cropped area.
• Measure the dimensions and the length of the diagonal of the cropped area.
• Write the first four terms of an arithmetic sequence that has the length of the
diagonal of the cropped area as its first term. Using the terms of your sequence
as diagonal lengths, find the four corresponding photo widths. What do you
notice about this list of widths?
• Write the first four terms of an geometric sequence that has the length of the
diagonal of the cropped area as its first term. Using the terms of your sequence
as diagonal lengths, find the four corresponding photo widths. What do you
notice about this list of widths?
Check students’ work.
Finishing the Project
The answers to the activities should help you complete your project. Prepare a
presentation or demonstration that summarizes how an artist, a designer, or a
photographer uses sequences. Present this information to your classmates. Then
discuss the sequences you made.
Reflect and Revise
Review your summary. Are your drawings clear and correct? Are your sequences
accurate? Practice your presentation in front of at least two people before
presenting it to the class. Ask for their suggestions for improvement.
Extending the Project
Geometric and arithmetic patterns are used in other aspects of design and in
other careers. Research other areas where sequences are applied.
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Chapter 9 Project Manager: Get the Picture
Getting Started
Read the project. As you work on the project, you will need a calculator, a metric
ruler, at least two types of graph paper, and materials on which you can record
your calculations. Keep all of your work for the project in a folder.
Checklist
Suggestions
☐ Activity 1: relating perspective and
arithmetic sequences
☐ Use art books from the school library or the
Internet.
☐ Activity 2: relating dimensions and
geometric sequences
☐ Use grid paper to draw simple geometric
designs.
☐ Activity 3: relating photo-cropping and
sequences
☐ Measure directly or use proportions to find
the widths.
☐ presentation
☐ Does your display include examples of
both arithmetic and geometric sequences?
What artists or work of art with which you
are familiar best demonstrate the concepts
of one-point perspective, two-point
perspective, or vanishing points?
Scoring Rubric
4
Calculations, sequences, and formulas are correct. Drawings are neat,
accurate, and clearly show the sequences. Explanations are thorough and
well thought out.
3
Calculations, sequences, and formulas are mostly correct with some minor
errors. Drawings are neat and mostly accurate. Explanations lack detail or are
not completely accurate.
2
Calculations contain both minor and major errors. Drawings are not
accurate.
1
Major concepts are misunderstood. Project satisfies few of the requirements
and shows poor organization and effort.
0
Major elements of the project are incomplete or missing.
Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.
Teacher’s Evaluation of the Project
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