answers - openstudyassets.openstudy.com/updates/attachments/5307cd93e... · answers know 2. . plan...

Prentice Hall Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

67

A N S W E R S


2

9-1 Think About a Plan Mathematical Patterns

Geometry Suppose you are stacking boxes in levels that form squares. Th e numbers of boxes in successive levels form a sequence. Th e fi gure 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 fi rst four levels?

Level 1: z z Level 2: z z Level 3: z z Level 4: z z

2. How many boxes of equal size would you need for the next lower level?

3. What is a recursive or explicit formula that describes the number of boxes in the nth level?

4. How many boxes would you need to add three levels?

z z 1 z z 1 z z 5 z z

5. What is a recursive or explicit formula that describes the total number of boxes in a stack of n levels?

6. How can you use your formula to fi nd the number of levels you will have with a stack of 285 boxes?

.

7. Suppose you are stacking a total of 285 boxes. Use your formula to fi nd how many levels you will have. Show your work.

8. You need z z levels to make a stack of 285 boxes.

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

1

9

25 36 49 110

4 9 16

25

explicit: an 5 n2

recursive: a1 5 1; an 5 an21 1 n2

Use the formula to fi nd the number of boxes in a stack with successive levels until

an L 285

PED-HSM11A2TR-08-1103-009-L01.indd 2 3/25/09 7:22:51 PM

9-1 ELL SupportMathematical Patterns

Choose the word or phrase from the list that best matches each sentence.

1. an ordered list of numbers

2. a formula that describes the nth term of a sequence using the number n

3. each number in a sequence

4. a formula that describes the nth term of a sequence by referring to preceding terms

Choose the word or phrase from the list that best completes each sentence.

5. For a sequence that is described by a recursive formula, the fi rst term in the

sequence is the .

6. In the sequence 2, 4, 6, 8, the number 4 is the second in the sequence.

7. Th e position of a term in a sequence can be represented by using

a(n) .

8. Th e formula an 5 3n 1 2 is a(n) .

9. An ordered list of numbers is called a(n) .

10. Th e formula an11 5 an 1 5 is a(n) .

explicit formula recursive formula sequence term

explicit formula initial condition recursive formula

sequence subscript number term

sequence

explicit formula

recursive formula

subscript number

recursive formula

explicit formula

sequence

term

initial condition

term


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Prentice Hall Gold Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

4

Determine whether each formula is explicit or recursive. Th en fi nd the fi rst fi ve terms of each sequence.

38. an 513 n 39. an 5 n2 2 6 40. a1 5 5, an 5 3an21 2 7

41. an 512

(n 2 1) 42. a1 5 5, an 5 3 2 an21 43. a1 5 24, an 5 2an21

44. Error Analysis Your friend says the explicit formula for the sequence 1, 8, 27, 64 is an 5 n2. Is she correct? Explain.

45. Writing Explain how to fi nd an explicit formula for a sequence.

46. Th e fi rst fi gure of a fractal contains one segment. For each successive fi gure, six segments replace each segment.

a. How many segments are in each of the fi rst four fi gures of the sequence? b. Write a recursive defi nition for the sequence.

47. Th e 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). b. Write an explicit formula for the measure of one exterior angle in a regular

polygon with n angles. c. Why would this formula not be meaningful for n 5 1 or n 5 2?

48. Reasoning In order to fi nd a term in a sequence, its position in the sequence is doubled and then two is added. What are the fi rst ten terms in the sequence?

49. Writing Explain the diff erence between a recursive and an explicit formula.

50. Open-Ended Write fi ve terms in a sequence. Describe the sequence using a recursive or explicit formula.

9-1 Practice (continued) Form G

Mathematical Patterns

explicit; 13, 23, 1, 43, 53

explicit; 0, 12, 1, 1 12, 2

explicit; 25, 22, 3, 10, 19

recursive; 5, 22, 5, 22, 5

recursive; 5, 8, 17, 44, 125

recursive; 24, 28, 216, 232, 264

She is incorrect; in order to fi nd each term

Look for a pattern in the

in the sequence, the term number must be cubed, not squared.

sequence and fi nd a mathematical rule that gives the nth term, given the number n.

An explicit formula defi nes how to fi nd the nth term directly from the number n, while a recursive formula defi nes how to fi nd each term from the previous term(s).

4, 6, 8, 10, 12, 14, 16, 18, 20, 22

Check students’ work.

No polygon has one or two angles.

an 5 360n

608

1, 6, 36, 216an 5 6an21 where a1 5 1

PED-HSM11A2TR-08-1103-009-L01.indd Sec1:4 3/25/09 7:22:57 PM

page 4

Find the fi rst six terms of each sequence.

1. an 5 22n 1 1 2. an 5 n2 2 1 3. an 5 2n2 1 1

4. an 5 1n 1 1 5. an 5 2n 1 2 6. an 5 2n2 2 n

7. an 5 4n 1 n2 8. an 513 n3 9. an 5 (22)n

Write a recursive defi nition for each sequence.

10. 214, 28, 22, 4, 10, c 11. 6, 5.7, 5.4, 5.1, 4.8, c 12. 1, 22, 4, 28, 16, c

13. 1, 3, 9, 27, c 14. 1, 12, 14, 18, 116, c 15. 2

3, 1, 113, 12

3, 2, c

16. 36, 39, 42, 45, 48, c 17. 36, 30, 24, 18, 12, c 18. 9.6, 4.8, 2.4, 1.2, 0.6, c

Write an explicit formula for each sequence. Find the twentieth term.

19. 7, 14, 21, 28, 35, c 20. 2, 8, 14, 20, 26, c 21. 5, 6, 7, 8, 9, c

22. 21, 0, 1, 2, 3, c 23. 3, 5, 7, 9, 11, c 24. 0.8, 1.6, 2.4, 3.2, 4, c

25. 14, 12, 34, 1, 54, c 26. 1

2, 14, 16, 18, 110, c 27. 2

3, 123, 22

3, 323, 42

3, c

Find the eighth term of each sequence.

28. 1, 3, 5, 7, 9, c 29. 400, 200, 100, 50, 25, c 30. 0, 22, 24, 26, 28, c

31. 1, 2, 4, 8, 16, c 32. 44, 39, 34, 29, 24, c 33. 0.7, 0.8, 0.9, 1.0, 1.1, c

34. 4, 11, 18, 25, 32, c 35. 114, 2

12, 5, 10, 20, c 36. 26, 29, 212, 215, 218,

c

37. A man swims 1.5 mi on Monday, 1.6 mi on Tuesday, 1.8 mi on Wednesday, 2.1 mi on Th ursday, and 2.5 mi on Friday. If the pattern continues, how many miles will he swim on Saturday?

9-1 Practice Form G


21, 23, 25, 27, 29, 211

15

128

53

an 5 7n; 140

an 5 n 2 2; 18

an 5 n4; 5

an 5 an21 1 6 where a1 5 214

an 5 3an21 where a1 5 1

an 5 an21 1 3 where a1 5 36

2, 2, 2, 2, 2, 2

5, 12, 21, 32, 45, 60

0, 3, 8, 15, 24, 35

3.125

9

160

an 5 6n 2 4; 116

an 5 2n 1 1; 41

an 5 12n; 1

40

an 5 an21 2 0.3 where a1 5 6

an 5 12 an21 where a1 5 1

an 5 an21 2 6 where a1 5 36

4, 6, 10, 18, 34, 66

13, 83, 9, 64

3 , 1253 , 72

3, 9, 19, 33, 51, 73

214

1.4

227

an 5 n 1 4; 24

an 5 0.8n; 16

an 5 n 2 13; 19

23


an 5 an21 1 13 where a1 5 2

3

an 5 12 an21 where a1 5 9.6

1, 6, 15, 28, 45, 66

22, 4, 28, 16, 232, 64

3.0 mi



68

A N S W E R S

Write an explicit formula for each sequence. Th en fi nd the tenth term.

13. 7, 10, 13, 16, c 14. 8, 9, 10, 11, 12, c 15. 212, 0, 12, 1, 1

12, c

an 5 3n 1 4

a10 5 3(10) 1 4 5 z z

16. 1, 4, 9, 16, c 17. 3, 1, 21, 23, 25, c 18. 1, 7, 25, 79, 241

19. Reasoning You and your friend are trying to fi nd the 80th term in the sequence 8, 14, 20, 26, 32, c. You use a recursive defi nition and your friend uses an explicit formula. Who will fi nd the 80th term fi rst? Why?

20. Your neighbor recently began learning to play the guitar. On the fi rst 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?

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?

9-1 Practice (continued) Form K


an 5 n2; 100

1.75 h

$78.20

Your friend will fi nd the 80th term fi rst 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.

34

an 5 22n 1 5; 215

an 5 n 1 7; 17

an 5 3n 2 2; 59,047

an 5 12n 2 1; 4


9-1 Practice Form K


Find the fi rst fi ve terms of each sequence.

1. an 5 4n 2 1 2. an 5 n2 1 4

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.

Th e fi rst fi ve terms are 3, 7, z z, z z, and z z.

3. a 512

n 1 2 4. an 5 3n 5. an 5 26n2

6. Write an explicit formula for a sequence with 3, 5, 7, 9, and 11 as its fi rst fi ve terms.


7. 2, 6, 12, 20, c 8. 120, 60, 30, 15, c

Identify the initial condition.

a1 5 2

Use n to express the relationship between successive terms.

9. 3, 8, 13, 18, c 10. 1, 3, 9, 27, c 11. 2, 3, 8, 63, c

12. Writing Explain the diff erence between a recursive defi nition and an explicit formula.

2.5, 3, 3.5, 4, 4.5

a1 5 3; an11 5 an 1 5

A recursive formula defi nes a sequence by the relationship between successive terms.An explicit formula describes the nth term of a sequence using the number n.

an 5 2n 1 1

an 5 an21 1 2n

a1 5 120; an11 5 Q12Ran

3, 9, 27, 81, 243

a1 5 1; an11 5 3an

11 15 19

5, 8, 13, 20, 29

26, 224, 254, 296, 2150

a1 5 2; an11 5 an2 2 1


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8

You can defi ne the terms in a sequence using an explicit formula or a recursive defi nition. You can use another method, called iteration, to form a sequence. Th e 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 fi rst term of a sequence be 0. What is f (0)? Let f (0) be the second term of the sequence. Write the sequence.

2. To create more terms of this sequence through iteration, continue to apply f (x) to each output. Th e third term in this sequence can be described as f ( f (0)). What is the third term?

3. Determine the fi rst 10 terms of this sequence. You already have the fi rst 3 terms.

4. Determine the fi rst 5 terms of the sequence formed through iterations off (x) 5

x2 1 1. Begin with x 5 2. Describe the sequence.

5. Will you get the same type of sequence if you start with a diff erent number?

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 diff erence between each pair of integers in the set. Th is produces a new set of three integers. Continue this process on each new set of three integers. Describe what eventually happens.

7. You can form fractals through iterations. Fractals are geometric fi gures just like circles or rectangles, but fractals have a special property that these geometric fi gures do not. You make fractals by iterating the fi gure 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 fi rst two iterations of a fractal called the Koch snowfl ake.

9-1 Enrichment Mathematical Patterns

0, 1

f (f (0)) 5 f (1) 5 6

0; 1; 6; 31; 156; 781; 3906; 19,531; 97,656; 488,281

2, 2, 2, 2, 2; all of the terms in the sequence are 2.

No; for

No matter what three integers you

example, if you start with x 5 0, the sequence is 0, 1, 1.5, c

choose to start with, the set will eventually repeat itself in combinations of the set {0, a, a}, where a is a positive integer.


page 8

Multiple Choice

For Exercises 1−6, choose the correct letter.

1. What are the fi rst fi ve terms of the sequence?

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. Th e formula an 5 3n 1 2 best represents which sequence?

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?

21, 0, 5, 12, 21 4, 7, 12, 19, 28 1, 4, 9, 16, 25 22, 1, 6, 13, 22

4. Th e sequence 4, 16, 36, 64, 100, ccan best be represented by which formula?

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?

3198 3200 4000 16,000

6. A student sets up a savings plan to transfer money from his checking account to his savings account. Th e fi rst 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?

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. Th e 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.

9-1 Standardized Test Prep Mathematical Patterns

C

G

D

G

A

G

[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



69

A N S W E R S

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 11 15 20 Spread the numbers in the sequence apart, leaving space between numbers.

12 13 14 15 Beneath each space, write what can be done to get the next number in the sequence.

In each term, the number that is added Find a pattern.to the previous term increases by one.

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. 5, 6, 8, 11, 15 2. 3, 6, 12, 24, 48 3. 1, 22, 4, 28, 16, 232

4. 1, 3, 9, 27, 81 5. 100, 95, 90, 85, 80 6. 15, 18, 21, 24, 27

7. 5, 25, 125, 625, 3125 8. 50, 49, 47, 44, 40 9. 240, 120, 60, 30, 15

10. 3, 5, 9, 15, 23 11. 280, 120, 2180, 270, 2405 12. 1, 5, 13, 29, 61

9-1 ReteachingMathematical Patterns

1. Each term is increased by one more than the previous term; 20, 26, 332. Each term is multiplied by 2 to get the next term; 96, 192, 3843. Each term is multiplied by 22 to get the next term; 64, 2128, 256

4. Each term is multiplied by 3 to get the next term; 243, 729, 21875. Each term is decreased by 5 to get the next term; 75, 70, 656. Each term is increased by 3 to get to the next term; 30, 33, 36

7. Each term is multiplied by 5 to get the next term; 15,625; 78,125; 390,6258. Each term is decreased by one more than the previous term; 35, 29, 229. Each term is divided by 2 to get the next term; 7.5, 3.75, 1.875

10. In each term, the number is increased by two more than the previous term; 33, 45, 5911. Each term is multiplied by 21.5 to get the next term; 607.5, 2911.25, 1366.87512. Each term is multiplied by 2 and then 3 is added to get the next term; 125, 253, 509


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page 10page 9

9-2 Think About a Plan Arithmetic Sequences

Transportation Suppose a trolley stops at a certain intersection every 14 min. Th e fi rst 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

1. If you defi ne 12:00 a.m. as minute 0, then 6:43 a.m. is z z from 0.

2. 8:15 a.m. is z z from 0 and 3:20 p.m. is z z from 0.

3. Th e trolley stops every z z.

Need

4. To solve the problem I need to fi nd:

.

Plan

5. What is an explicit formula for the number of minutes after 12:00 a.m. that the trolley gets to the stop?

6. Use your formula to fi nd the smallest n that gives the minutes just after 8:15 a.m. that the trolley arrives at the stop.

7. Using this n in your formula, when does the trolley stop? How long do you have to wait for this trolley?

8. Use your formula to fi nd the smallest n that gives the minutes just after 3:20 p.m. that the trolley arrives at the stop.

9. Using this n in your formula, when does the trolley stop?How long do you have to wait for this trolley?

an 5 403 1 (n 2 1)14

403 min

495 min 920 min

14 min

8

at 501 min

6 min

38

at 921 min1 min

the closest times that the trolley gets to the stop that are after 8:15 A.M.

and 3:20 P.M.


page 12

To fi nd a recursive defi nition for a sequence, you compare each term to the previous term.

Problem

What is the recursive defi nition for the sequence?

800, 2400, 200, 2100, 50, c

To fi nd the recursive defi nition for a sequence, fi rst describe the sequence in words.

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 fi nd the next term, divide the previous term by 22.

Exercises


13. 38, 33, 28, 23, c 14. 7, 14, 28, 56, c 15. 25, 27, 29, 211, c

16. 2, 6, 18, 54, c 17. 4.5, 5, 5.5, 6, c 18. 17, 20, 24, 29, c

9-1 Reteaching (continued) Mathematical Patterns

800, 2400, 200, 2100, 50, g

The initial term is 800. The terms are alternatively negative g

g

and positive.

The next term in thesequence will be 225.

To find the next term in the sequence,divide by negative two.

an 5 an21 2 5 where a1 5 38



an 5 an21 1 0.5 where a1 5 4.5

an 5 an21 2 2 where a1 5 25

an 5 an21 1 (n 1 1) where a1 5 17


Arithmetic Sequence

An arithmetic sequence is a sequence where the diff erence 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

2. 3, 9, 15, 21, 27, c

3. 0, 15, 30, 45, 60, c

4. 0, 1, 3, 6, 10, c

Use the formula an 5 a 1 (n 2 1)d to fi nd the indicated term in each arithmetic sequence.

5. Find the 12th term in the sequence that begins 3, 6, 9, c



Arithmetic Mean

Th e arithmetic mean is the average of a set of numbers. Th e arithmetic mean of two numbers x and y is found using the formula displayed below.

x 1 y2

Sample Th e arithmetic mean of 4 and 6 is 4 1 62 5

102 5 5.

Find the missing number in the arithmetic sequence. Th is number is the arithmetic mean of the two given numbers.

8. c, 13, , 37, c

9. c, 26, , 42, c

10. c, 45, , 99, c

9-2 ELL Support Arithmetic Sequences

not arithmetic

arithmetic

arithmetic

not arithmetic

36

226

417

25

34

72



70

A N S W E R S

9-2 Practice Form K

Arithmetic Sequences

Determine whether each sequence is arithmetic. If so, identify the common diff erence.

1. 1, 4, 7, 10, c 2. 6, 10, 14, 18, 22, c 3. 1, 3, 6, 10, 15, c

4 2 1 5 3

7 2 4 5 3

10 2 7 5 3

Th is sequence is arithmetic.

Th e common diff erence is z z.

4. 216, 213, 29, 24, 2, c 5. 2, 9, 16, 23, 30, c 6. 43, 56, 69, 82, c

7. Reasoning Is the sequence represented by the formula an 5 4n 1 8 arithmetic? Explain.

Find the 24th term of each arithmetic sequence.

8. 4, 6, 8, 10, 12, c 9. 2, 5, 8, 11, 14, c 10. 9, 5, 1, 23, 27, 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

a24 5 4 1 46

a24 5 z z

Find the missing terms in the following arithmetic sequences.

11. 2, ___, ___, 14, c 12. 3, z z, z z, 21, c 13. 65, z z, z z, 32, c

14 5 2 1 3d

12 5 3d

d 5 4

2 1 4 5 z z6 1 4 5 z z

14. Error Analysis Noah used the formula an 5 a 1 (n 2 1)d to fi nd the 12th term in the sequence 2, 4, 7, 11, 16, c. Did Noah fi nd the correct term? How do you know?

not arithmetic

Yes; the difference between consecutive terms is 4.

3

50

6

10

9 15 54 43

arithmetic; 7

arithmetic; 4

71

sequence that is not arithmetic.No; Noah applied the explicit formula for arithmetic sequences to a

arithmetic; 13

not arithmetic

283



14

Find the arithmetic mean an of the given terms.

35. an21 5 5, an11 5 11 36. an21 5 17, an11 5 3

37. an21 5 28, an11 5 29 38. an21 5 20.6, an11 5 3.8

39. an21 5 y 2 z, an11 5 y 40. an21 5 2t 1 3, an11 5 4t 2 1

41. Open-Ended Write an arithmetic sequence of at least fi ve terms with a positive common diff erence.

42. Error Analysis On your homework, you write that the missing term in the arithmetic sequence 31, ___, 41, c is 351

2. Your friend says the missing term is 36. Who is correct? What mistake was made?

43. Reasoning Explain why 84 is the missing term in the sequence 89, 86.5, ___, 81.5, c.

44. Writing Describe the general process of fi nding a missing term in an arithmetic sequence.

45. You are making an arrangement of cubes in concentric rings for a sculpture. Th e number of cubes in each ring follows the pattern below.

1, 9, 17, 25, 33, c

a. Is this an arithmetic sequence? Explain. b. What are the next three terms? c. If the sequence continues to the 100th term in this pattern, what will that term be?

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. Th is 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.

b. Write the fi rst fi ve terms of the sequence. c. Write an explicit formula for the number of people the organization expects

to serve each year. d. How many people would the organization expect to serve in the 20th year?



108

28.5 1.6

y 2 z2 3t 1 1

a fi ve-term sequence with a positive commondifference

Your friend is correct. You did not takethe average of 31 and 41 correctly to fi nd the missing term of 36.

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.

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.

Yes; there is a common difference of 8.41, 49, 57

793

an 5 an21 1 5 where a1 5 3232, 37, 42, 47, 52

an 5 32 1 5(n 2 1)127 people



13

9-2 Practice Form G



1. 2, 3, 5, 8, c 2. 0, 23, 26, 29, c

3. 0.9, 0.5, 0.1, 20.3, c 4. 3, 8, 13, 18, . . .

5. 14, 215, 244, 273, c 6. 3.2, 3.5, 3.8, 4.1, c

7. 234, 228, 222, 216, c 8. 2.3, 2.5, 2.7, 2.9, c

9. 127, 140, 153, 166, c 10. 11, 13, 17, 25, c

Find the 43rd term of each sequence.

11. 12, 14, 16, 18, c 12. 13.1, 3.1, 26.9, 216.9, c

13. 19.5, 19.9, 20.3, 20.7, c 14. 27, 24, 21, 18, c

15. 2, 13, 24, 35, c 16. 21, 15, 9, 3, . . .

17. 1.3, 1.4, 1.5, 1.6, c 18. 22.1, 22.3, 22.5, 22.7, c

19. 45, 48, 51, 54, c 20. 20.073, 20.081, 20.089, c

Find the missing term of each arithmetic sequence.

21. c 23, 7 , 49, c 22. 14, 7 , 28, c 23. c 29, 7 , 33, c

24. c 14, 7 , 15, c 25. c 245, 7 , 239, c 26. c 25, 7 , 22, c

27. 22, 7 , 2, c 28. c 26, 7 , 2, c 29. 234, 7 , 77, c

30. c 245, 7 , 212, c 31. 22, 7 , 456, c 32. c 34, 7 , 345, c

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?

34. Writing Explain how you know that the sequence 109, 105, 101, 97, 93, cis arithmetic.

no yes; 23

yes; 20.4 yes; 5

yes; 229 yes; 0.3

yes; 6 yes; 0.2

yes; 13

96

36.3

464

5.5

171

no

2406.9

299

2231

210.5

20.409

The sequence has a common difference between terms of 24.

40 acres

189.5

21.5

−3.5

3121

242

22

227228.5

0

14.5

36


page 15

page 14page 13

Prentice Hall Foundations Algebra 2 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

16


15. c4, ___, 18, c 16. c9, z z , 37, c

Find the arithmetic mean of the given terms.

4 1 18 5 22

22 4 2 5 11

Th e missing term is z z .

17. 46, z z , 28, c 18. 212, z z , 24, c 19. c4, z z , 244, c

20. Error Analysis Your friend used the arithmetic mean to fi nd 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?

21. An architect is designing a building with sides in the shape of a trapezoid. Th e number of windows on each fl oor forms an arithmetic sequence. Th ere are 124 windows on the fi rst fl oor and 116 windows on the second fl oor.

a. Write an explicit formula to represent the sequence. b. How many windows are on the tenth fl oor?

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?

23. Th ere is a puddle 1.4 cm deep in your backyard. After one minute of rain, the puddle was 1.45 cm deep. Th e 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?



11

23

37

He subtracted 3 from 29 when he should have added 3 and 29; 16

28

an 5 132 2 8n52 windows

$688

3.65 cm

220


page 16


71

A N S W E R S

Multiple Choice


1. Which sequence is an arithmetic sequence?

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?

76 80 84 99

3. What are the missing terms of the arithmetic sequence 5, __, __, 62, c?

19, 24 19, 34 24, 43 43, 62

4. What is the missing term of the arithmetic sequence 25, __, 45, c?

30 35 37 40

5. Th e seventh and ninth terms of an arithmetic sequence are 197 and 173. What is the eighth term?

161 180 185 221

6. An artist is creating a tile mosaic. She uses 4 green tiles in the fi rst 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?

32 58 134 137

Extended Response

7. What is the 100th term in the arithmetic sequence beginning with 3, 19, c? Show your work.

9-2 Standardized Test Prep Arithmetic Sequences

[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

A

I

C

G

C

I


page 19

page 17 page 18

Th ere are many types of sequences. One interesting type of sequence is the Farey sequence. Th e fi rst four Farey sequences are:

F1: e 01 , 1

1 f F2: e 0

1 , 12 , 1

1 f F3: e 0

1 , 13 , 1

2 , 23 ,1

1 f F4: e 0

1 , 14 , 1

3 , 12 , 2

3 , 34 , 1

1 fEach 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?

Th e 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.

Th e fractions 05 and 55 will not be added because they simplify to 01 and 11. Insert the

fractions 15 , 25 , 3

5, and 45 in the Farey sequence F4.

F5: e 01 , 1

5 , 14 , 1

3 , 25 , 1

2 , 35 , 2

3 , 34 , 4

5 , 11 f

Exercises

1. How many terms are in each of the fi rst fi ve Farey sequences?

2. What are the terms for the Farey sequence F6?

3. What will be the new terms in the Farey sequence F7?

4. Since 11 is a prime number, how many more terms will be in the sequence F11 compared to the sequence F10?

5. Is there any limit to how large n can be?

6. Can you give examples of any other sequences?

9-2 EnrichmentArithmetic Sequences

2, 3, 5, 7, 11

e 01, 16, 15, 14, 13, 25, 12, 35, 23, 34, 45, 56, 11 f

e 17, 27, 37, 47, 57, and 67 f

No, n can be any positive integer although the computations become tedious.

Answers may vary. Sample: arithmetic, geometric, and Fibonacci

10

PED-HSM11A2TR-08-1103-009-L02.indd Sec2:18 3/25/09 7:25:07 PMpage 20


20

9-2 Reteaching (continued) Arithmetic Sequences

To solve word problems that involve arithmetic sequences, identify the common diff erence 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 fi xed 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 fi nd 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?

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?

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?

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?

$110

$143

31 rows

105 min


9-2 ReteachingArithmetic Sequences

Th e 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 diff erence.

• 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 fi rst three terms are 20, 16.5, and 13.

20 2 16.5 5 3.5 First, fi nd the common difference. The difference between 16.5 2 13 5 3.5 consecutive terms is 3.5. The sequence decreases. The common

difference is 23.5.

an 5 a 1 (n 2 1) d Use the explicit formula.

a15 5 20 1 (15 2 1)(23.5) 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 fi nd a15.

Exercises

Find the 25th term of each sequence.

1. 20, 18, 16, 14, c 2. 0.0057, 0.0060, 0.0063, c

3. 4, 0, 24, 28, c 4. 0.2, 0.7, 1.2, 1.7, c

5. −10, 28.8, 27.6, 26.4, c 6. 22, 26, 30, 34, c

228

292

18.8

0.0129

12.2

118



72

A N S W E R S

9-3 Think About a Plan Geometric Sequences

Athletics During your fi rst 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?

.

2. Is the sequence arithmetic, geometric, or neither?

3. What is the fi rst term of the sequence?

4. What is the common ratio of the sequence?

5. What is the problem asking you to determine?

Planning the Solution

6. Write a formula for the sequence.

Getting an Answer

7. Evaluate your formula to fi nd the number of miles you run during your twelfth week of training.

Answers may vary. Sample: Start with 10, and multiply it and each successive term

by 120% or 1.2

the 12th term of a geometric sequence that represents the number of miles you run

each week

geometric

10

1.2

an 5 10(1.2)n21

about 74.3



9-3 ELL SupportGeometric Sequences

Use the chart below to review vocabulary. Th ese 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 fi rst) to its preceding term is a constant value.

Th e 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

Th e common ratio in the sequence 1, 4, 16, 64, 256, . . . is 4.

Geometric mean Th e geometric mean of two

numbers x and y is !xy .

Th e geometric mean of the numbers 4 and 9 is !4 ? 9 5 !36 5 6.

1. Th e terms in the sequence 2, 6, 18, 54, 162, . . . all share a with their preceding terms.

2. Th e numbers 8 and 2 have a of 4.

3. Th e consecutive terms in a all share a common ratio.

Identify each sequence as arithmetic or geometric.

4. 2, 8, 32, 128, . . .

5. 1, 3, 9, 27, . . .

6. 1, 4, 7, 10, . . .

Identify the common ratio for each geometric sequence.

7. 3, 12, 48, 192, . . .

8. 12, 60, 300, 1500, . . .

Find the missing term in the geometric sequence.

9. . . . , 4, ___, 16, . . .

10. . . . , 9, ___, 25, . . .

common ratio

geometric mean

geometric sequence

geometric

geometric

arithmetic

4

5

15

8


page 23

page 22page 21


24

Identify each sequence as arithmetic, geometric, or neither. Th en fi nd the next two terms.

32. 9, 3, 1, 13, . . . 33. 1, 0, 22, 25, . . . 34. 2, 22, 2, 22, . . .

35. 23, 2, 7, 12, . . . 36. 1, 22, 25, 28, . . . 37. 1, 22, 3, 24, . . .

Write an explicit formula for each sequence. Th en generate the fi rst fi ve terms.

38. a1 5 3, r 5 22 39. a1 5 5, r 5 3 40. a1 5 21, r 5 4

41. a1 5 22, r 5 23 42. a1 5 32, r 5 20.5 43. a1 5 2187, r 5 13

44. a1 5 9, r 5 2 45. a1 5 24, r 5 4 46. a1 5 0.1, r 5 22

47. Th e deer population in an area is increasing. Th is 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.

b. Find the expected deer population for the fourth year of the sequence.

48. You enlarge the dimensions of a picture to 150% several times. After the fi rst increase, the picture is 1 in. wide.

a. Write an explicit formula to model the width after each increase. b. How wide is the photo after the 2nd increase? c. How wide is the photo after the 3rd increase? d. How wide is the photo after the 12th increase?

Find the missing terms of each geometric sequence. (Hint: Th e geometric mean of positive fi rst and fi fth terms is the third term. Some terms might be negative.)

49. 12, j , j , j , 0.75 50. 29, j , j , j , 22304

For the geometric sequence 6, 18, 54, 162, . . . , fi nd the indicated term.

51. 6th term 52. 19th term 53. nth term


Geometric Sequences

geometric; 19, 127

an 5 3(22)n21; 3, 26, 12, 224, 48

an 5 5(3)n21; 5, 15, 45, 135, 405

an 5 32(20.5)n21; 32, 216, 8, 24, 2

an 5 9(2)n21; 9, 18, 36, 72, 144

an 5 24(4)n21; 24, 216, 264, 2256, 21024

an 5 0.1(22)n21; 0.1, 20.2, 0.4, 20.8, 1.6

an 5 21(4)n21; 21, 24, 216, 264, 2256

arithmetic; 17, 22

neither; 29, 214

arithmetic; 211, 214

geometric; 2, 21

neither; 5, 26

an 5 2537(1.025)n21

about 2732

an 5 1(1.5)n21

1.5 in.2.25 in.about 86.5 in.

an 5 22(23)n21; 22, 6, 218, 54, 2162

6, 3, 1.5 or 26, 3, 21.5

1458 2,324,522,934 6(3)n21

236, 2144, 2576 or 36, 2144, 576

an 5 2187Q13Rn21; 2187, 729, 243, 81, 27


page 24

9-3 Practice Form G

Geometric Sequences

Determine whether each sequence is geometric. If so, fi nd the common ratio.

1. 3, 9, 27, 81, . . . 2. 4, 8, 16, 32, . . . 3. 4, 8, 12, 16, . . .

4. 4, 28, 16, 232, . . . 5. 1, 0.5, 0.25, 0.125, . . . 6. 100, 30, 9, 2.7, . . .

7. 25, 0, 5, 10, . . . 8. 64, 232, 16, 28, . . . 9. 1, 4, 9, 16, . . .

Find the tenth term of each geometric sequence.

10. 2, 4, 8, . . . 11. 1, 3, 9, . . . 12. 22, 6, 218, . . .

13. 23, 9, 227, . . . 14. 23, 212, 248, . . . 15. 25, 25, 2125, . . .

16. 13,

19,

127, . . . 17. 0.3, 0.6, 1.2, . . . 18. 1

4, 12, 1, . . .

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. b. 8th arc is 27 mm; 10th arc is 3 mm c. 5th arc is 25 cm; 7th arc is 1 cm d. 100th arc is 18 ft; 98th arc is 2 ft

Find the missing term of each geometric sequence. It could be the geometric mean or its opposite.

20. 4, j , 16, . . . 21. 9, j , 16, . . . 22. 2, j , 8, . . .

23. 3, j , 12, . . . 24. 2, j , 50, . . . 25. 4, j , 5.76, . . .

26. 625, j , 25, . . . 27. 13, j , 3, . . . 28. 0.5, j , 0.125, . . .

29. Writing Explain how you know that the sequence 400, 200, 100, 50 is geometric.

30. Open-Ended Write a geometric sequence of at least seven terms.

31. Error Analysis A student says that the geometric sequence 30, __, 120 can be completed with 90. Is she correct? Explain.

yes; 3

yes; 22

yes; 2

yes; 0.5

yes; 20.5

no

no

yes; 0.3

no

1024

59,049

159,049

19,683

2786,432

153.6

about 19.2 in.

128

9 mm5 cm6 ft

68

66

6125

The sequence has a common ratio of 12 or 0.5 between terms.

any seven-term sequence with a common ratio

No; the sequence can be completed with 60 with a common ratio of 2.

61

610

612 64

64.8

60.25

39,366

9,765,625



73

A N S W E R S


25

9-3 Practice Form K

Geometric Sequences

Determine whether each sequence is geometric. If so, fi nd the common ratio.

1. 1, 3, 9, 27, c 2. 2, 5, 8, 11, 14, c

Find the ratios between consecutive terms.31 5

93 5

279

Th e sequence is geometric.

Th e common ratio is z z.

3. 22, 24, 28, 216, c 4. 500, 50, 5, 0.5, c 5. 0, 25, 50, 75, 100, c

6. Open-Ended Write a geometric sequence with a common ratio of 14. Explain how you developed the sequence.

Find the ninth term of each geometric sequence.

7. 3, 12, 48, 192, c 8. 2, 6, 18, 54, c 9. 1875, 375, 75, 15, c

Use the explicit formula.

an 5 a1 ? rn21

a9 5 3(48)

a9 5 3(65,536)

a9 5 z z

Find the missing terms of each geometric sequence.

10. 2, ___, ___, 128, c 11. 1, z z, z z, 8, c 12. 108, z z, z z, 4, c

Identify the common ratio.

an 5 a1 ? rn21

a4 5 2r421

128 5 2r3

64 5 r3

4 5 r

Th e second term is z z.Th e third term is z z.

3

196,608

8

32

2 364 12

geometric; 2

Answers may vary. Sample: 64, 16, 4, 1, . . . . I divided the fi rst term by 4 to get the second term. Then I divided the second and third terms by 4.

geometric; 110

13,122

not geometric

0.0048

not geometric


page 27

page 26page 25


28

9-3 EnrichmentGeometric Sequences

Doubling Periods in Geometric Sequences

Consider the geometric sequence 3, 4 12, 6

34, 10

18, . . . .

1. Describe how the terms of the sequence are related.

2. For any term of the sequence, how many terms does it take before the value of the term has at least doubled?

Th e 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?

3. Write the fi rst ten terms of the geometric sequence a1 5 3, r 5 1.1 to two decimal places.

4. What is the doubling period for a1 5 3? for a2 5 3.3?

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?

Th e idea of a doubling period applies to certain everyday situations. For example, under optimum conditions, bacteria reproduce by splitting in two. Th eir 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?

6. If a count is taken every 40 min, how many terms are in the sequence? What is the common ratio? What is the doubling period?

7. In both cases, how many hours does it take the bacteria to double?

Each term is 112 times the preceding term.

2

2 terms

3, 3.3, 3.63, 3.99, 4.39, 4.83, 5.31, 5.85, 6.43, 7.07

8 terms; 8 terms

1 R | r | R 2

7; !2; 2 terms

10; 3"2; 3 terms

2


page 28

Find the missing term of each geometric sequence. It could be the geometric mean or its opposite.

13. 5, ___, 45, c 14. 2, z z, 72, c

Find the geometric mean of 5 and 45.

!xy

!45 ? 5

!225

z z

15. 14, z z, 2 14, c 16. 175, z z, 7, c 17. 1.2, z z, 43.2, c

18. Error Analysis On a recent math test, your classmate was asked to fi nd the missing term in the geometric sequence 4, ___, 256. Her answer was 130. What error did your classmate make? What is the correct answer?

19. Th e 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 defi nition to represent this sequence. b. If this pattern continues, what will be the bacteria population after 4 h?

20. A corporation earned a profi t of $420,000 in its fi rst year of operation. Over the next 10 years, the company’s CEO hopes to increase the profi t by 8% each year. If the CEO reaches her goal, what will be the company’s profi t in its seventh year, to the nearest dollar?


Geometric Sequences

She found the arithmetic mean of 256 and 4 rather than the geometric mean; 32

an 5 14 ? 2n21

3584

$666,487

w12

w15

234 w35 w7.2


Multiple Choice


1. What is the 10th term of the geometric sequence 1, 4, 16, . . .?

40 180,224 262,144 2,883,584

2. Which sequence is a geometric sequence?

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, . . .?

25 50 75 100

4. What could be the missing term of the geometric sequence 212, __,234, . . .?

24 26.375 3 4

5. In the explicit formula for the 9th term of the geometric sequence 1, 6, 36, . . . what number is 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?

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.

9-3 Standardized Test PrepGeometric Sequences

C

H

A

H

A

I

[2] 3.125; an 5 ar n21; an 5 100Q12Rn21; a6 5 100Q12R5 5 3.125;

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



74

A N S W E R S


30

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. Th is means a 5 1000. You also know that her rent in 2003 was $1092.73. Th is 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 fi nd the common ratio r in order to fi nd the rent in 2009, a10.

Step 3 Use the explicit formula to fi nd r.

an 5 arn 2 1 Write the explicit formula.

1092.73 5 (1000)r4 2 1 Substitute a 5 1000, a4 5 1092.73, and n 5 4.

1092.73 5 1000r3 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 fi nd the rent in 2009, a10.

an 5 arn 2 1 Write the explicit formula.

a10 5 (1000)(1.03)1021 Substitute a 5 1000, r 5 1.03, and n 5 10.

a10 5 (1000)(1.03)9 Simplify.

a10 < 1304.77 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 fi rst day, and 12.1 mi on the third day, how much will she bike on the fi fth day? By what percent does she increase the amount she bikes each day?

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?

24. From 2005 to 2009, a teen raised her babysitting rates by a fi xed 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?

9-3 Reteaching (continued)

Geometric Sequences

14.641 mi; 10%

$341.28

$12.59/h; 12%



29

9-3 ReteachingGeometric Sequences

• A geometric sequence has a constant ratio between consecutive terms. Th is 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, . . .

r 5155 5

4515 5 3 Find r by calculating the common ratio between consecutive terms. This is a

geometric sequence because there is a common ratio between consecutive terms.

an 5 5(3)n21 Substitute a 5 5 and r 5 3 into the explicit formula to fi nd a formula for the nth term of the sequence.

a12 5 5(3)11 Substitute n 5 12 to fi nd the 12th term of the sequence.

a12 5 885,735 Remember to fi rst calculate 311, then multiply by 5.

Exercises

Find the indicated term of the geometric sequence.

1. 4, 2, 1, . . . Find a10. 2. 5, 152 , 45

4 , . . . Find a8. 3. 6, 22, 23, . . . Find a12.

4. 1, 2 23, 49, . . . Find a7. 5. 100, 200, 400, . . . Find a9. 6. 8, 32, 128, . . . Find a4.

Write the explicit formula for each sequence. Th en generate the fi rst fi ve terms.

7. a1 5 1, r 512 8. a1 5 2, r 5 3 9. a1 5 12, r 5 3

10. a1 5 1, r 514 11. a1 5 5, r 5

110 12. a1 5 1, r 5

13

13. a1 5 5, r 5 2 14. a1 5 1, r 5 3 15. a1 5 3, r 5 6

16. a1 5 3, r 5 3 17. a1 5 2, r 5 2 18. a1 5 2, r 512

19. a1 5 1, r 515 20. a1 5 3, r 5 4 21. a1 5 5, r 5

14

1128

64729

an 5 1Q12Rn21; 1, 12, 14, 18, 116

an 5 5(2)n21; 5, 10, 20, 40, 80

an 5 3(3)n 2 1; 3, 9, 27, 81, 243 an 5 2Q12Rn21; 2, 1, 12, 14, 18

an 5 5Q14Rn21; 5, 54, 516, 5

64, 5256an 5 1Q15Rn21; 1, 15, 1

25, 1125, 1

625 an 5 3(4)n 2 1; 3, 12, 48, 192, 768

an 5 2(2)n 2 1; 2, 4, 8, 16, 32

an 5 1(3)n 2 1; 1, 3, 9, 27, 81 an 5 3(6)n 2 1; 3, 18, 108, 648, 3888

an 5 1Q14Rn21; 1, 14, 116, 1

64, 1256 an 5 5Q 1

10Rn21; 5, 12, 120, 1

200, 12000 an 5 1Q13Rn21; 1, 13, 19, 1

27, 181

an 5 2(3)n 2 1; 2, 6, 18, 54, 162 an 5 12(3)n 2 1; 12, 36, 108, 324, 972

10,935128

25,600

2 259,049

512


page 31

page 30page 29

9-4 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. Th e fi rst 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.

2. What are a1 and d for the sequence that represents the number of seats in each row?

a1 5 z z d 5 z z

3. Write an explicit formula for the arithmetic sequence that represents the number of seats in each row.

4. Write an arithmetic series to represent the number of seats in the theater.

5. How can you use a graphing calculator to evaluate the series?

.

6. Find the total seating capacity of the theater.

7. Write a series for the number of seats in each set of 5 rows.

8. Use your graphing calculator to evaluate each series.

9. What are the ticket prices for each set of 5 rows?

10. What is the total amount of money generated by a full house?

an 5 a1 1 (n 2 1)d

an 5 3n 1 15

a5

n51(3n 1 15); a

10

n56(3n 1 15); a

15

n511(3n 1 15); a

20

n516(3n 1 15)

930

$46,950

120; 195; 270; 345

$60; $55; $50; $45

a20

n51

(3n 1 15)

18 3

Answers may vary. Sample: Use the sum command and the sequence command

sum(seq(3N115,N,1,20))


page 32

9-4 ELL SupportArithmetic Series

For Exercises 1−5, draw a line from each word or phrase in Column A to its defi nition in Column B.

Column A Column B

1. arithmetic series A. a series that continues without end

2. fi nite series B. the sum of the terms of a sequence

3. infi nite 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 fi rst and a last term

Identify the following series as fi nite or infi nite.

6. 2 1 6 1 18 1 54

7. 3 1 10 1 17 1 24 1 c

8. 2 1 10 1 50 1 250 1 c

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 fi lling in the upper and lower limits.

3 1 11 1 19 1 27 1 35 1 c 1 115 5 au

u(8n 2 5)

fi nite

infi nite

infi nite

lower limit: n 5 1; upper limit: 15



75

A N S W E R S


33

9-4 Practice Form G

Arithmetic Series

Find the sum of each fi nite arithmetic series.

1. 1 1 3 1 5 1 7 1 9 2. 5 1 8 1 11 1 c 1 26

3. 4 1 9 1 14 1 c 1 44 4. (210) 1 (225) 1 (240) 1 c 1 (285)

5. 17 1 25 1 33 1 c 1 65 6. 125 1 126 1 127 1 c 1 131

7. A bookshelf has 7 shelves of diff erent widths. Each shelf is narrower than the shelf below it. Th e bottom three shelves are 36 in., 31 in., and 26 in. wide.

a. Th e shelf widths decrease by the same amount from bottom to top. What is the width of the top shelf?

b. What is the total shelf space of all seven shelves?

Write each arithmetic series in summation notation.

8. 4 1 8 1 12 1 16 9. 10 1 7 1 4 1 c 1 (25)

10. 1 1 3 1 5 1 c 1 13 11. 3 1 7 1 11 1 c 1 31

12. (220) 1 (225) 1 (230) 1 c 1 (265) 13. 15 1 25 1 35 1 c 1 75

Find the sum of each fi nite series.

14. a4

n51(n 2 1) 15. a

6

n52(2n 2 1) 16. a

8

n53(n 1 25)

17. a5

n52(5n 1 3) 18. a

4

n51(2n 1 0.5) 19. a

6

n51(3 2 n)

20. a10

n55n 21. a

4

n51(2n 2 3) 22. a

6

n53(3n 1 2)

Use a graphing calculator to fi nd the sum of each series.

23. a15

n51(n 1 3) 24. a

12

n51(2n 2 1) 25. a

20

n512n2

26. a25

n51(n3 1 2n) 27. a

50

n51(n2 2 4n) 28. a

25

n55(5n3 1 3n)

25

6 in.147 in.

6

165

106,275 37,825 528,570

144 5740

35 183

82 22 23

45 222 62

a4

n514n

a7

n51(2n 2 1)

a10

n51(25n 2 15) a

7

n51(10n 1 5)

a8

n51(4n 2 1)

a6

n51(23n 1 13)

216

287 896

2285

124

PED-HSM11A2TR-08-1103-009-L04.indd Sec1:33 3/25/09 8:26:49 PMpage 35

page 34page 33


36


13. 3 1 8 1 13 1 c1 268

Find an explicit formula Find the value of n for 268. Write the summation notation.for the nth term. 268 5 5n 2 2

an 5 a1 1 (n 2 1)d 270 5 5n a

n51

Qz zRan 5 3 1 (n 2 1)5 54 5 n

an 5 5n 2 2

14. 1 1 7 1 13 1 c1 343 15. 5 1 7 1 9 1 c1 131

16. Tabitha used tiles to make the design shown at the right. Th e fi rst column has 2 tiles, the second column has 4 tiles, and the pattern continues.

a. Write an explicit formula for the sequence. b. Write the summation notation for a related series with 24 tiles

in the 12th column. c. How many tiles are in the design if there are a total of 12 columns?

17. Your brother is preparing for basketball season. He shot 26 baskets on the fi rst 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? b. How many baskets will he have shot during those 30 days?


Arithmetic Series

u5n 2 2

54

an 5 2n

156 tiles

200 baskets3390 baskets

a58

n51

(6n 2 5) a64

n51

(2n 1 3)

a12

n51

(2n)


page 36

Determine whether each list is a sequence or a series and fi nite or infi nite.

29. 7, 12, 17, 22, 27 30. 3 1 5 1 7 1 9 1 c 31. 8, 8.2, 8.4, 8.6, 8.8, 9.0, c

32. 1 1 5 1 9 1 c 1 21 33. 40, 20, 10, 5, 2.5, 1.25, c 34. 10 1 20 1 30 1 40 1 50

35. An embroidery pattern calls for fi ve stitches in the fi rst row and for three more stitches in each successive row. Th e 25th row, which is the last row, has 77 stitches. Find the total number of stitches in the pattern.

36. A marching band formation consists of 6 rows. Th e fi rst 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?

37. Writing Explain how you can identify the diff erence between a series and a sequence.

38. a. Open-Ended Write three explicit formulas for arithmetic sequences. b. Write the fi rst seven terms of each related series. c. Use summation notation to rewrite the series. d. Evaluate each series.

39. Error Analysis A student identifi es the series 10 1 15 1 20 1 25 1 30 as an infi nite arithmetic series. Is he correct? Explain.

40. Mental Math Use mental math to evaluate a3

1(2n 1 1).

41. To train new employees, an employer off ers a bonus after 30 work days as follows. An employee must turn in one report on the fi rst 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?


Arithmetic Series

sequence; fi nite

series; fi nite

A series is the sum of terms in a sequence, which is indicated by summation notation or addition signs.

Check students’ work. Sequences should be arithmetic and contain seven terms.

No; the series is a fi nite arithmetic series. An infi nite arithmetic series would continue indefi nitely.

sequence; infi nite series; fi nite

series; infi nite sequence; infi nite

1025 stitches

19 musicians; 84 musicians

15

900


9-4 Practice Form K

Arithmetic Series

Identify each list as a series or a sequence and fi nite or infi nite.

1. 2, 6, 10, 14, c 2. 1 1 4 1 7 1 10 1 13 3. 4, 10, 16, 22, 28

4. 5 1 12 1 19 1 26 1 33 5. 1.4 1 1.1 1 0.8 1 0.5 1 . . . 6. 22 2 11 2 20 2 29 2 c


7. 1 1 3 1 5 1 c1 99 8. 3 1 7 1 11 1 15 1 c1 55

Find the number of terms. Find the sum. Find the number of terms. Find the sum.

an 5 a1 1 (n 2 1)d Sn 5n2 (a1 1 an) an 5 a1 1 (n 2 1)d Sn 5

n2 (a1 1 an)

99 5 1 1 (n 2 1)2 S50 5502 (1 1 99)

99 5 1 1 2n 2 2 5 25(100)

100 5 2n 5 z z 50 5 n

9. 106 1 101 1 96 1 c1 1 10. 2 1 10 1 18 1 c1 378

11. (24) 1 (29) 1 (214) 1 c1 (299)

12. Reasoning Is it possible to fi nd the sum of an infi nite arithmetic series? Explain.

infi nite sequence

fi nite series

1177

fi nite series

infi nite series

9120

2500

406

fi nite sequence

infi nite series

21030

No; an infi nite arithmetic series has a never-ending number of terms, so it is impossible to add them all.



76

A N S W E R S

9-4 Standardized Test PrepArithmetic Series

Multiple Choice

For Exercises 1–6, choose the correct letter.

1. What is the sum of the odd integers 1 to 99?

2450 2500 2550 4950

2. Which of the following is an infi nite series?

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. Th e high school choir is participating in a fundraising sales contest. Th e choir will receive a bonus if they make 20 sales in their fi rst week and improve their sales by 3 in every subsequent week. What is the minimum number of sales the choir could make in the fi rst 12 weeks to qualify for the bonus?

13 53 438 5015

4. What is summation notation for the series 5 1 7 1 9 1 c 1 105?

a51

n51(2n 1 3) a

51

n51(n 1 3) a

50

n51(2n 1 3) a

51

n57(n 1 3)

5. What is the upper limit of the summation a100

n51(n 2 2)?

1 2 98 100

6. What is the sum of the series a30

n51(2n 1 2)?

62 66 990 1980

Short Response

7. What is the sum of the fi nite arithmetic series 2 1 4 1 6 1 c 1 50? Show your work.

B

H

C

F

D

H

[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


page 39

page 38page 37

page 40

9-4 Reteaching Arithmetic Series

Summation notation shows the upper limit, lower limit, and explicit formula for the terms of a series.

To fi nd the sum of an arithmetic series written in summation notation:

• list the terms and add them, or use the formula Sn 5n2(a1 1 an)

Problem

What is the sum of the series written in summation notation?

a. a4

n52(5 2 2n)

a4

n52(5 2 2n) Circle the upper and lower limits. Box the

explicit formula.

n 5 2 n 5 3 n 5 4 In circles, write all possible values of n, from the lower limit to the upper limit.

n 5 2 n 5 3 n 5 4

5 2 2(2) 5 2 2(3) 5 2 2(4) Under each circle copy the explicit formula, substituting the value in the circle for n.

a4

n52(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.

5 1 1 (21) 1 (23) Evaluate each expression.

5 23 Find the sum of the terms.

Th e sum of the series is 23.

b. a15

n51(4n 2 1)

Use the formula Sn 5n2(a1 1 an). First, fi nd n, a1, and an. The upper limit is 15.

a1 5 4(1) 2 1 5 3 Evaluate the explicit formula at n 5 1.

an 5 a15 5 4(15) 2 1 5 59 Evaluate the explicit formula at n 5 15.

Sn 5152 (3 1 59) Substitute n 5 15, a1 5 3, and an 5 59.

5 465 Simplify.

Th e sum of the series is 465.

ExercisesFind the sum of each fi nite series.

1. a3

n51(n 2 4) 2. a

4

n51

13 n 3. a

8

n53(3n 2 1) 4. a

8

n53

2n3

5. a9

n53(4 2 2n) 6. a

5

n518n 7. a

7

n524n 8. a

7

n51(3 2 2n)

26

108 235256 120

93103 22


9-4 Reteaching (continued) Arithmetic Series

Problem

Th e debate club is off ering a prize at the end of 10 weeks to a current member who brings three new members for the fi rst meeting, and then increases the number of new members they bring each week by two thereafter. One member qualifi ed 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 fi rst meeting, so a 5 3.

A member must also bring two more new members to each meeting, so d 5 2.

Th e contest extends for 10 weeks, so n 5 10.

Step 2 Identify the information you are trying to fi nd.

You want to fi nd the 10th term, a10, and the sum of the fi rst 10 terms, S10.

Step 3 Use the explicit formula to fi nd a10.

an 5 a 1 (n 2 1)d Write the explicit formula. a10 5 3 1 (10 2 1)2 Substitute a 5 3, d 5 2, and n 5 10. a10 5 21 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 fi nd the total number of new members brought by the winner.

Sn 5n2(a1 1 an) Write the formula for the sum of an arithmetic series.

S10 5102 (3 1 21) Substitute a1 5 3, a10 5 21, and n 5 10.

S10 5 120 Simplify.

Th e debate club had 120 new members brought in by the winner of the contest.

Exercises

9. Th e seating arrangement for a recital uses 20 seats in the fi rst 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 fi rst nine rows?

10. With the help of a tutor, a student’s weekly quiz scores have increased during the fi rst 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 fi rst eight scores?

34 seats; 36 seats; 252 seats

95; 100; 660


9-4 EnrichmentArithmetic 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 fi nite series whose sum you are trying to fi nd. 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 fi rst equation and the left side of the second equation?

2. What is the sum of each vertically-aligned pair of quantities on the right side of the equal signs?

3. How may such pairs are there?

4. Because each pair has the same sum, use multiplication to express the sum of all the pairs on the right side.

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.

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 diff erence of d.

6. What is the nth term, in terms of a1, d, and n?

7. Write the sum S of the n terms of the series, where each number is written in terms of a1 and d. Th en write the sum in reverse order, lining up terms.

8. What is the sum of each vertical pair of quantities on the right side?

9. How many such pairs are there?

10. Express the sum of all the pairs using multiplication.

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.

12. Show that your equation is equivalent to S 5 n2(a1 1 an). Hint: Use your answer to Exercise 6.

2S

101

100

100 3 101 5 10,100

2S 5 10,100; S 5 5050

a1 1 (n 2 1)d

2a1 1 (n 2 1)dn

nf2a1 1 (n 2 1)dg

2S 5 nf2a1 1 (n 2 1)dg; S 5nf2a1 1 (n 2 1)dg

2

S 5nf2a1 1 (n 2 1)dg

2

5nfa1 1 a1 1 (n 2 1)dg

2

5nfa1 1 fa1 1 (n 2 1)dgg

2

5nfa1 1 ang

2 5 n2

(a1 1 an)

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



77

A N S W E R S


43

9-5 Practice Form G

Geometric Series

Evaluate each fi nite series for the specifi ed number of terms.

1. 40 1 20 1 10 1 c; n 5 10 2. 4 1 12 1 36 1 c; n 5 15

3. 15 1 12 1 9.6 1 c; n 5 40 4. 27 1 9 1 3 1 c; n 5 100

5. 0.2 1 0.02 1 0.002 1 c; n 5 8 6. 100 1 200 1 400 1 c; n 5 6

7. Th is 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?

Determine whether each infi nite geometric series diverges or converges. State whether each series has a sum.

8. 3 132 1

34 1 c 9. 4 1 2 1 1 1 c 10. 17 1 15.3 1 13.77 1 c

11. 6 1 11.4 1 21.66 1 c 12. 220 2 8 2 3.2 2 c 13. 50 1 70 1 98 1 c

Evaluate each infi nite geometric series.

14. 8 1 4 1 2 1 1 1 c 15. 1 113 1

19 1

127 1 . . .

16. 120 1 96 1 76.8 1 61.44 1 c 17. 1000 1 750 1 562.5 1 421.875 1 c

18. Suppose your business made a profi t of $5500 the fi rst year. If the profi t increased 20% per year, fi nd the total profi t over the fi rst 5 yr.

19. Th e end of a pendulum travels 50 cm on its fi rst swing. Each swing after the fi rst, it travels 99% as far as the preceding swing. How far will the pendulum travel before it stops?

20. A seashell has chambers that are each 0.82 times the length of the enclosing chamber. Th e outer chamber is 32 mm around. Find the total length of the shell’s spiraled chambers.

21. Th e fi rst year a toy manufacturer introduces a new toy, its sales total $495,000. Th e company expects its sales to drop 10% each succeeding year. Find the total expected sales in the fi rst 6 years. Find the total expected sales if the company off ers the toy for sale for as long as anyone buys it.

79.921875 28,697,812

about 74.99 about 40.5

0.22222222 6300

$8553.71

converges; yes converges; yes converges; yes

converges; yes diverges; nodiverges; no

16 1.5

600 4000

$40,928.80

5000 cm

about 177.78 mm

$2,319,367.05; $4,950,000


9-5 ELL Support Geometric Series

Problem

What is the sum of the geometric series 2 1 6 1 18 1 54 1 c1 1458?

62 5

186 5

5418 5 3 Identify the common ratio and the nth term.

nth term 5 1458

an 5 a1rn21 Use the explicit formula.

1458 5 2 ? 3n21 Substitute 2 for a1, 3 for r, and 1458 for an.

729 5 3n21 Divide each side by 2.

729 is 36, so n 2 1 5 6 and n 5 7 Use a calculator.

Sn 5a1(1 2 rn)

1 2 r Use the sum formula.

S7 52(1 2 37)

1 2 3 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?

41 5164 5

6416 5 4 .

nth term 5 1024

an 5 a1rn21 .

1024 5 1 ? 4n21 .

1024 5 4n21 .

1024 is 45, so n 2 1 5 5 and n 5 6 .

Sn 5a1(1 2 rn)

1 2 r .

S6 51(1 2 46)

1 2 4 .

S6 5 1365

Identify the common ratio and the nth term

Use the explicit formula

Substitute 1 for a1, 4 for r, and 1024 for an

Divide each side by 1

Use a calculator

Use the sum formula

Substitute 1 for a1, 4 for r, and 6 for n


page 43

page 42page 41

page 44

9-5 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. Th en each of these people calls three others, and so on.

a. Make a diagram to show the fi rst 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 fi rst six stages.

c. How many employees have been notifi ed after stage six?

1. What type of diagram can you make to represent the telephone chain?

2. Make a diagram to show the fi rst three stages in the telephone chain.

3. What expression represents the number of calls made at stage n?

4. Write the series that represents the total number of calls made through the fi rst six stages.

5. What is the sum of this series?

6. Write the sum formula.

7. Use the sum formula to fi nd how many employees have been notifi ed after stage six.

8. Does your answer agree with your sum from Exercise 5?

tree diagram

3n

1092

yes

Sn 5a1(1 2 rn)

1 2 r

Sn 5a1(1 2 rn)

1 2 r 53(1 2 36)

1 2 3 5 1092

3 1 9 1 27 1 81 1 243 1 729


Determine whether each series is arithmetic or geometric. Th en evaluate the series for the specifi ed number of terms.

22. 2 1 5 1 8 1 11 1 c; n 5 9 23. 18 1

116 1

132 1

164 1 c; n 5 8

24. 23 1 6 2 12 1 24 2 c; n 5 10 25. 22 1 2 1 6 1 10 1 c; n 5 12

26. 4 1 8 1 16 1 32 1 c; n 5 15 27. 5 1 10 1 15 1 20 1 c; n 5 20

Evaluate each infi nite series that has a sum.

28. a`

n51

5Q23Rn21 29. a

`

n51

(22. 1)n21 30. a`

n51

Q212Rn21 31. a

`

n51

2Q53Rn21

32. Open Ended Write an infi nite geometric series that converges to 2. Show your work.

Find the specifi ed value for each infi nite geometric series.

33. a1 5 5, S 5253 , fi nd r 34. S 5 108, r 5 13, fi nd a1

35. a1 5 3, S 5 12, fi nd r 36. S 5 840, r 5 0.5, fi nd a1

37. Error Analysis Your friend says that an infi nite geometric series cannot have a sum because it’s infi nite. You say that it is possible for an infi nite geometric series to have a sum. Who is correct? Explain.

38. Writing Describe in general terms how you would fi nd the sum of a fi nite geometric series.


Geometric Series

You are; an infi nite geometric series with »r… less than 1 has a series of partial sums that converges towards a number.

Identify the fi rst term, common ratio, and nth term. Use the explicit formula to fi nd n. Then, use the sum formula with the fi rst term, common ratio, and n to fi nd the sum of the series.

arithmetic; 126 geometric; 2551024

geometric; 1023 arithmetic; 240

geometric; 131,068 arithmetic; 1050

15 no sum 23

no sum

25

72

0.75 420




78

A N S W E R S

Gridded Response

Solve each exercise and enter your answer in the grid provided.

1. What is the value of a1 in the series a20

n50

3Q12R

n?

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 fi rst stage, one person calls fi ve families. In the second stage, each of the fi ve families calls another fi ve 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 fi nite geometric series

a5

n50

8Q14Rn

?

5. What is the sum of the geometric series 1 113 1

19 1 c? Enter your answer

as a fraction.

Answers

1. 2. 3. 4. 5.

9-5 Standardized Test Prep Geometric Series

9

76543210

9876543210

987654

210

9876543210

9876543210

9876543210

–

3

89

76543210

9876543210

987654

210

9876543210

9876543210

9876543210

–

3

89

76543210

9876543210

987654

210

9876543210

9876543210

9876543210

–

3

89

76543210

9876543210

987654

210

9876543210

9876543210

9876543210

–

3

89

76543210

9876543210

987654

210

9876543210

9876543210

9876543210

–

3

8

3

3

7 2 8

7

2

8

4

4

1 1

1 1

3 2/

32


page 45

page 47

page 46


48

9-5 EnrichmentGeometric Series

An infi nite geometric series converges if the absolute value of the common ratio is less than 1 ( u r u , 1). A power series is an infi nite series where each term depends on a variable x. Each value of x will give you a specifi c infi nite series, which may converge or diverge.

1. Evaluate the expression 11 2 x for x 5 2 and for x 5

12.

2. You can evaluate many expressions with a power series called a Taylor series.

To evaluate 11 2 x , you use the Taylor series 1 1 x 1 x2 1 x3 1 c. What is

this infi nite series written in summation notation?

3. Determine the sum of the fi rst fi ve terms of the Taylor series

1 1 x 1 x2 1 x3 1 cfor x 5 2 and for x 5 12.

4. For which value of x is your computation above a better approximation for the

value of the expression 112x? 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?

5. You can use a diff erent Taylor series to evaluate ex. Write the Taylor series

1 1x1! 1

x2

2! 1x3

3! 1 cin summation notation.

6. Evaluate ex for x 512. Evaluate the fi rst four terms of the Taylor series

1 1x1! 1

x2

2! 1x3

3! 1 cfor x 512. Round your answers to the nearest

thousandth.

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.

21; 2

a`

n50

xn

31; 1.9375

12;»x… R 1

a`

n50

xn

n!

1.649; 1.646

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.


page 48

Determine whether each infi nite geometric series diverges or converges. Find the sum if the series converges.

8. 1 114 1

116 1 c 9. 2 1 8 1 32 1 c

Because ur u 5 P 14 P , 1, the series converges.

S 5a1

1 2 r 51

1 2 14

5 z z

10. 12 1

116 1

1128 1 c 11. 1

4 138 1

916 1 c 12. 2 2 25 1 2

25 2 c

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?

14. Your math teacher asks you to choose between two off ers. Th e fi rst off er is to receive one penny on the fi rst day, 3 pennies on the second day, 9 pennies on the third day, and so on, for 14 days. Th e second off er is to receive 4 pennies on the fi rst day, 8 pennies on the second day, 16 pennies on the third day, and so on, for 14 days. Which off er is better? What is the diff erence between the total amounts received?


Geometric Series

134

5

converges; 47diverges converges; 12

3

diverges

113

about 25,833 min

the fi rst offer; 2,325,952 pennies


9-5 Practice Form K

Geometric Series

Find the sum of each fi nite geometric series.

1. 2 1 6 1 18 1 c1 4374 2. 1 1 2 1 4 1 c1 2048

Find the number of terms. Use the sum formula.

an 5 a1rn21 Sn 5 a1(1 2 rn)1 2 r

4374 5 2 ? 3n21 S8 52(1 2 38)

1 2 3

2187 5 3n21 5 z z 37 5 2187

n 5 8

3. 8 1 4 1 2 1 c11

256 4. 3 1 9 1 27 1 . . . 1 6561

5. 24 2 8 2 16 2 c2 2048

6. Find the sum of the geometric series 2 2 4 1 8 2 16 1 c1 8192. Explain how you found the sum.

7. A family farm produced 2400 ears of corn in its fi rst 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?

N 16

24092

48,729

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.

9840

6560

4095



79

A N S W E R S

page 49

page 51

page 50

page 52


49

9-5 ReteachingGeometric Series

• Th e sum of a fi nite geometric series is Sn 5a1(1 2 rn)

1 2 r , where a1 is the fi rst term, r is the common ratio, and n is the number of terms.

• Th e sum of an infi nite geometric series with u r u , 1 is S 5a1

1 2 r , where a1 is the fi rst 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 fi rst ten terms of the geometric series 8 1 16 1 32 1 64 1 128 1 c?

a1 5 8 a1 is the fi rst term in the series.

r 5168 5

3216 5

6432 5

12864 5 2 Simplify the ratio formed by any two consecutive terms to fi nd r.

n 5 10 n is the number of terms in the series to be added together.

S10 58(1 2 210)

1 2 2 Substitute a1 5 8, r 5 2, and n 5 10 into the formula for the sum of a fi nite geometric series.

58(21023)

21 Simplify inside the parentheses.

5 8184 Simplify.

ExercisesEvaluate the fi nite series for the specifi ed number of terms.

1. 3 1 12 1 48 1 192 1 c; n 5 6 2. 8 1 2 112 1

18 1 c; n 5 5

3. 210 2 5 2 2.5 2 1.25 2 c; n 5 7 4. 10 1 (25) 152 1 Q25

4R 1 c; n 5 11


5. 10 1 5 1 2.5 1 c 6. 21 12

11 24

121 1 c 7. 14 1

732 1

49256 1 c

8. 12 2

15 1

225 2 c 9. 2

16 1

112 2

124 1 c 10. 20 1 16 1

645 1 c

11. 12 1 4 143 1 c 12. 1

4 218 1

116 2 c 13. 2

3 12

15 12

75 1 c

4095

20

514

18 16

56

219 100

21113 2

263532

3415512

34132


9-5 Reteaching (continued)

Geometric Series

Problem

Your neighbor hosts a family reunion every year. In 2000, it costs $1500 to host the reunion. Th eir 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?

Th e cost is a geometric sequence that decreases by the same percent each year.

an 5 arn21 Write the explicit formula.

an 5 (1500)(0.90)n21 Substitute a1 5 1500, r 5 1 2 0.10 5 0.90 in the explicit formula.

To fi nd the cost of the reunion in 2005 (n 5 6), substitute values into the explicit formula.

an 5 arn21 Write the explicit formula.

an 5 (1500)(0.90)621 Substitute a1 5 1500, r 5 0.90, n 5 6 in the formula.

an < 886 Simplify.

Th e cost of hosting the reunion in 2005 was $886.

To fi nd the total of hosting the reunions from 2000 to 2009, a10

n51(1500)(0.90)n21,

fi nd the sum of the geometric series.

Sn 5a1(1 2 rn)

1 2 r Write the formula for the sum of a geometric series.

S10 51500(1 2 0.9010)

1 2 0.90 Substitute a1 5 1500, r 5 0.90, n 5 10 in the formula.

S10 < 9770 Simplify.

Th e cost of hosting the reunions from 2000 to 2009 was $9770.

Exercise

14. In 1990, a vacation package cost $400. Th e cost has increased 10% per year. a. What are the values of a1 and r? b. What is a rule for the cost of the vacation? c. What was cost of the vacation in 1995? d. What was the total cost of the vacations from 1990 to 1999? e. If the pattern continued until 2009, what was the total cost of the vacations?

a1 5 400, r 5 1.1an 5 (400)(1.10)n21

$644.20$6374.97

$22,910


Chapter 9 Quiz 1 Form G

Lessons 9-1 through 9-2

Do you know HOW?


1. an 5 n2 1 2n 2. an 5 n 2 6

Find the seventh term of each sequence.

3. 10, 9, 7, 4, c 4. 2, 4, 8, 16, c


5. 13, 19, 25, 31, c 6. 16, 24, 36, 54, c


7. 4, ___ , 24, 34, c 8. 100, ___, 92, c

Do you UNDERSTAND?

9. Vocabulary Explain what it means for a formula to be an explicit formula.

10. Open-Ended Give an example of an arithmetic sequence.

3, 8, 15, 24, 35

211

yes; d 5 6

14

An explicit formula describes the nth term in a sequence using n.


96

no

128

25, 24, 23, 22, 21

PED-HSM11A2TR-08-1103-009-Quiz.indd 51 3/25/09 7:37:34 PM

Chapter 9 Quiz 2 Form G

Lessons 9-3 through 9-5

Do you know HOW?

Find the eighth term of each geometric sequence.

1. 4, 8, 16, c 2. 20,480; 5120; 1280; c

Find the seventh term of each sequence.

3. 2, 4, 8, 14, c 4. 1, 23, 9, 227, c

Determine whether each sequence is arithmetic or geometric. Th en evaluate the fi nite series for the specifi ed number of terms.

5. 1 1 3 1 9 1 c; n 5 8 6. 25 1 32 1 39 1 c; n 5 12


7. a`

n51Q21

2Rn21

8. a`

n513(0.4)n21

Do you UNDERSTAND?

9. Open-Ended Write an arithmetic series that has a negative sum.

10. Reasoning Can an infi nite geometric series converge when the common ratio is greater than 1? Explain. Give an example.

44

23

5

512

geometric; 3280

Answers may vary. Sample: 1.2 1 0.2 2 0.8 2 . . . 2 8.8 5 241.8

Answers may vary. Sample: No, the terms of the series grow in absolute value so they cannot have a fi nite sum; two possible series with r 5 2 are 3 1 6 1 12 1 24 1 . . . and (21) 1 (22) 1 (24) 1 . . . .

arithmetic; 762

729

54



80

A N S W E R S

Chapter 9 Quiz 1 Form K

Lessons 9–1 through 9–2

Do you know HOW?

Find the fi rst four terms of each sequence.

1. an 5 5n 2 2 2. an 5 n3 1 5 3. an 5 2 12 n


4. 80, 40, 20, 10, c 5. 4, 10, 16, 22, c 6. 3, 21, 147, 1029, c


7. 5, 9, 13, 17, c 8. 4, 1, 22, 25, c 9. 1.2, 1.6, 2, 2.4, c

Use the arithmetic mean to fi nd the missing term in each arithmetic sequence.

10. c6, z z, 28, c 11. c2, z z, 214, c 12. c1.4, z z, 6.8, c

Do you UNDERSTAND?

13. Writing Describe the diff erence between a recursive defi nition and an explicit defi nition of a sequence.

14. Tim takes the stairs up to his offi ce. He enters the ground fl oor of the building and climbs 12 steps to reach the fi rst fl oor. He climbs a total of 24 steps to reach the second fl oor and 36 steps to reach the third fl oor. How many steps will Tim climb to reach his offi ce on the 16th fl oor?

3, 8, 13, 18

a1 5 80; an11 5 12an

73

17

An explicit defi nition describes the nth term of a sequence using the number n.A recursive defi nition relates each term to the next.

192 steps

6, 13, 32, 69

a1 5 4; an11 5 an 1 6

247

26

212, 21, 23

2, 22

a1 5 3; an11 5 7an

8

4.1


page 53

page 55

page 54


Do you know HOW?

Find the eighth term of each geometric sequence.

1. 4, 12, 36, 108, c 2. 2, 1, 12, 14, c 3. 0.04, 0.2, 1, 5, c


4. 3 1 6 1 9 1 c1 72 5. 6 1 12 1 18 1 c1 108

6. (22) 1 (27) 1 (212) 1 c1 (2102)


7. 1 1 5 1 9 1 c1 85 8. 5 1 11 1 17 1 c1 371

9. 212 1 204 1 196 1 c1 (220)

Find the sum of each fi nite geometric series.

10. 5 1 15 1 45 1 c1 10,935 11. 16 1

112 1

124 1 c1 1

384

12. 1 2 4 1 16 2 c2 16,384

Do you UNDERSTAND?

13. A guitar-making company produced 60 guitars this month. Th e 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?

Chapter 9 Quiz 2 Form K

Lessons 9–3 through 9–5

8748

900

21092

a30

n51

(220 2 8n)

213,107

about 869 guitars

a22

n51

(4n 2 3)

16,400

1026

a62

n51

(6n 2 1)

about 0.33

N 0.0156 3125


page 56

Chapter 9 Chapter Test Form G

Do you know HOW?

Write a recursive defi nition and an explicit formula for each sequence. Th en fi nd a10.

1. 41, 46, 51, 61, c 2. 1, 10, 100, 1000, 10000, c 3. 3, 6, 12, 24, 48, c

Find the fi rst fi ve terms in each arithmetic sequence.

4. an 5 3n 1 2 5. an 5 n 1 5

6. an 5 12n 7. an 5 2n 1 10

Determine whether each sequence is arithmetic, geometric, or neither. Th en fi nd the ninth term.

8. 3, 12, 48, 192, c 9. 22, 27, 212, 217, c

10. 10, 2, 25, 225, c 11. 1

2, 2, 72, 5, c

Find the missing term of each geometric sequence. It could be its geometric mean or its opposite.

12. 16, 7, 4 13. 25, 7, 225

14. 2, 7, 50 15. 1, 7, 49

16. 34, 7, 3 17. 36, 7, 4

Find the sum of each fi nite series.

18. a5

n51(n 2 1) 19. a

8

n513n

20. a15

n51(3n 1 1) 21. a

20

n51(5 2 n)

an 5 an21 1 5 where a1 5 41; an 5 41 1 5(n 2 1) or an 5 36 1 5n; 86

an 5 10an21 where a1 5 1; an 5 1(10)n21; 1,000,000,000

an 5 2an21 where a1 5 3; an 5 3(2)n21; 1536

5, 8, 11, 14, 17

12, 24, 36, 48, 60

geometric; 196,608

6 8

10

375 2110

9840

6 10

6 32 6 12

6 7

6 75

geometric; 278,125

arithmetic; 242

arithmetic; 252

9, 8, 7, 6, 5

6, 7, 8, 9, 10


Chapter 9 Chapter Test (continued) Form G


22. 30 1 22.5 1 16.875 1 c 23. 15 2 3 1 0.6 2 0.12 1 c

24. 12 1 6 1 3 1 c 25. 4 2 2 1 1 212 1 c

26. 21312 1 9 2 6 1 c 27. 25 2

52 2

54 2 c

Determine whether each series is arithmetic or geometric. Th en evaluate the fi nite series for the specifi ed number of terms.

28. 5 1 9 1 13 1 17 1 c; n 5 10 29. 35 1 70 1 140 1 280 1 c; n 5 7

30. 6 1 (218) 1 54 1 (2162) 1 c; n 5 8 31. 8 1 11 1 14 1 17 1 c; n 5 6

32. 10 1 8 1 6 1 4 1 c; n 5 10 33. 20 1 4 145 1

425 1 c; n 5 6

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. b. Write a recursive defi nition to model the number of bulbs she plants each day. c. How many bulbs will the gardener plant on October 15? d. What is the total number of bulbs she plants from October 1 to October 15, inclusive?

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, fi nd the total number of concrete blocks needed to build the steps.

Do you UNDERSTAND?

36. Writing Explain why an infi nite geometric series with r 5 1 diverges. Include an example in your explanation.

37. Open-Ended Write a sequence and describe it using both an explicit defi nition and a recursive formula.

38. Reasoning What does a recursive defi nition have that an explicit formula does not? Explain.

12.5120

24

330 blocks

210

83

28 1

10

arithmetic; 230

an 5 20 1 3(n 2 1)an 5 an21 1 3 where a1 5 2062 bulbs

615 bulbs

geometric; 29840

arithmetic; 10 geometric; 15,624

625

arithmetic; 93

geometric; 4445

Answers may vary. Sample: a`

n515(1)n21 5 5 1 5 1 5 1 . . .;The series diverges

because the number 5 is added an infi nite number of times

Answers may vary. Sample: 30; 300; 3000; 30,000; 300,000; . . .; an 5 10an21 where a1 5 30; an 5 30(10)n21

Answers may vary. Sample: A recursive defi nition 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.



81

A N S W E R S

Do you know HOW?

Find the 9th term of each geometric sequence.

13. 5, 10, 20, 40, c 14. 32, 28, 2, 20.5, . . . 15. 22, 210, 250, 2250, c


16. 8 1 12 1 16 1 c1 116 17. (23) 1 (29) 1 (215) 1 c1 (2201)

18. 1 1 5 1 9 1 c1 157

Determine whether each infi nite geometric series diverges or converges. If the series converges, state the sum.

19. 4, 2, 1, 12 , . . . 20. 6, 18, 54, 162 21. 5, 21, 0.2, 20.04, . . .

Do you UNDERSTAND?

22. Error Analysis Your friend calculated the sum of the fi nite geometric series 2 1 8 1 32 1 c1 32,768. Her answer was 131,080. What error did she make? What is the correct sum?

23. Writing Find the possible values of the missing term in the following geometric sequence, and explain how you found the answer.

6, z z, 96, c

Chapter 9 Test (continued) Form K

1280

1736

3160

She used the formula for the sum of an arithmetic series rather than the sum of a geometric series; 43,690

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.

−3468

converges; 8

N 0.0005

diverges

−781,250

converges; 4.16

±24


Do you know HOW?


1. an 5 3n 1 4 2. an 5 n2 1 n 3. an 5 12 n 2 2

Write an explicit formula for each sequence. Th en fi nd the 12th term.

4. 2, 6, 12, 20, c 5. 2.5, 3, 3.5, 4, c 6. 2, 5, 10, 17, 26, c


7. 2, 5, 8, 11, c 8. 56, 50, 44, 38, c 9. 2.2, 2.6, 3, 3.4, c

Do you UNDERSTAND?

10. Writing Find the missing term in the sequence below. Th en explain how you found the term.

c12, z z, 44, c

11. Reasoning Rita must fi nd the 35th term in the sequence that begins 2, 9, 16, 23, c. She needs to fi nd the answer as fast as possible. Should Rita use a recursive defi nition or an explicit formula? Why?

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?

Chapter 9 Test Form K

7, 10, 13, 16, 19

an 5 n(n 1 1); 156

59

51 people

Answers may vary. Sample: First, I found the sum of 12 and 44, which is 56. Then I divided the sum by 2 to fi nd the missing term, 28.

recursive defi nition will require her to go through many iterations of the defi nition. If she uses an explicit formula, she will be able to substitute the value into the formula to fi nd the answer.

explicit formula; using a

2, 6, 12, 20, 30

an 5 12 n 1 2; 8

−58

28

−1.5, −1, −0.5, 0, 0.5

an 5 n2 1 1; 145

9.8


page 57

page 59

page 58

page 60

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. d. Write a recursive defi nition and an explicit formula for the sequence of function

values. e. Find three terms of the sequence between 512 and 8192, and identify these

as arithmetic or geometric means. Explain your reasoning.

Task 2 a. Determine whether the sequence 27, 9, 3, 1, . . . is geometric, arithmetic, or neither.

Justify your response. b. Write a recursive defi nition and an explicit formula for this sequence. c. Use summation notation to write the series related to the fi rst ten terms

of the sequence give in part (a). Th en evaluate this series. d. Use summation notation to write the series related to the infi nite

sequence given in part (a). Determine whether this series diverges or coverages. If the series converages, fi nd its sum.

e. Describe a real-world situation that can be modeled by this sequence.

[4] Student correctly uses calculator to view the function and constructs table of values using positive integers for x-values. Student correctly identifi es the sequence as geometric and justifi es answer. Student correctly writes recursive defi nition and explicit formula for the sequence, fi nds three terms, and identifi es these as geometric means with justifi cation.

[3] Student correctly completes parts (a), (b), and (c). Justifi cation may not be fully developed. Student completes parts (d) and (e) with only minor errors and some justifi cation.

[2] Student correctly completes parts (a), (b), and (c). Justifi cation is not given. Student writes recursive defi nition and explicit formula with one or more errors. Student fi nds three terms with one or more major errors. Justifi cation 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.

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.

an 5 2an21 where a1 5 2; an 5 2n

geometric; There is a common ratio of 2.

Y1 5 2ˆX

Y 5 32X 5 5

X248163264

123456X 5 0

Y1

geometric; There is a common ratio of 13. an 5 13 an21 where

a1 5 27; an 5 27Q13Rn21

a10

n5127Q13Rn21; 29,524

729

a`

n5127Q13Rn21; converges; 81

2

[4] Student correctly determines that the sequence is geometric. Student correctly fi nds a recursive defi nition and explicit formula for the sequence. Student correctly writes the series using summation notation, fi nds the sum of the fi rst ten terms of the series, determines that the infi nite 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 defi nitions 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.

PED-HSM11A2TR-08-1103-009-Quiz.indd Sec1:59 3/25/09 7:37:44 PM

Chapter 9 Performance Tasks (continued)

Task 3 a. Determine whether the sequence 2, 8, 14, 20, 26,c is arithmetic geometric, or neither.

Justify your response. b. Write a recursive defi nition and an explicit formula for this sequence. c. Find three terms of the sequence between 62 and 86, and identify

these as arithmetic or geometric means. Explain your reasoning.

d. Use summation notation to write the series related to the infi nite sequence given in part (a). Find the sum of the fi rst ten terms of the series.

e. Describe a real-world situation that can be modeled by the sequence given in part (a).

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. c. Draw and use inscribed rectangles 1 unit wide to approximate the area

under the curve for the given interval. 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.

Arithmetic; there is a common difference of 6.

Check student’s drawing.

13 units2

18 units2


an 5 6 1 an21 where a1 5 2; an 5 2 1 6(n 2 1)

68, 74, and 80; arithmeticmeans; 74 is the average of 62

and 86, 68 is the average of 62 and 74, and 80 is the average of 74 and 86.

a`

n51(6n 2 4); 290

[4] Student uses a calculator to view the function and constructs a table of values using positive integers for x-values. Student correctly identifi es the sequence as geometric and justifi es answer. Student writes a recursive defi nition and an explicit formula for the sequence, fi nds three terms, and identifi es these as arithmetic means with justifi cation. Student uses summation notation to write a series and correctly fi nds the sum of the fi rst 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.

10

25

23 3

[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 fi nd 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.



82

A N S W E R S


63

Chapter 9 Project Teacher Notes: Get the Picture

About the ProjectTh e Chapter Project gives students an opportunity to use sequences, explicit formulas, and recursive formulas to change the sizes of drawings and photos. Th ey 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: ResearchingStudents research perspective, create drawings in perspective, write arithmetic sequences, and determine explicit or recursive formulas for their sequences.

Activity 2: DesigningStudents use grid paper to enlarge designs. Th ey use the same ratios repeatedly to draw lengths which form geometric sequences. Th ey then write explicit or recursive formulas for their sequences.

Activity 3: AnalyzingStudents crop photos. Th en they enlarge the cropped portions, writing sequences for the widths of the enlargements.

Finishing the ProjectYou 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.


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.

11. How can the relationship between variables in the table be described?

12. Use the sequence 100, 95, 90, 85, . . . a. Describe the sequence in words. b. Find the next three terms.

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 fi nd the amount of water in the tank after 24 h.

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 fi rst? Use composite functions. Show your work.

x

1

2

4

5

y

20

10

5

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 fi ve units.

w 5 25t 1 10,000; 9880 gal

[4] $2; student defi nes both functions and subtracts one from the other correctly.[3] Student defi nes 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.

The variables in the table, x and y, have an inverse variation relationship. As x-values increase, y-values decrease, but the product of their pairs remains constant.

42 6 8�2�4

2468

�4

x

O

y

This is an arithmetic sequence in which each term is fi ve less than the previous one.

80, 75, 70



61

Chapter 9 Cumulative Review

Multiple Choice

1. What is the y-intercept of y 5 0.75(3)x?

(0, 0.75) (0, 2.25) (3, 0) (4, 0)

2. What is x2 1 8x 1 15

x2 2 x 2 12 in simplest form?

x 1 5x 1 3 x 1 5

x 2 4 x 1 5x 1 4

(x 1 5)(x 1 3)(x 2 4)(x 1 3)

3. Which expression is equivalent to "3 x2

"6 x2?

2!x 2"x23 !3 x "3 x2

4. How is the polynomial 2x2 2 x3 1 4x 1 17 classifi ed by degree?

linear quadratic cubic quartic

5. Th e discriminant of a quadratic equation has a value of 0. Which of the following is true?

Th ere is one real solution. Th ere is one complex solution.

Th ere are no real solutions. Th ere are two complex solutions.

6. Which of these does not have the same value as the others?

log28 log39 log464 log5125

7. Which inequality is graphed?

y # x 1 4 y $ x 1 4

y # x 2 4 y , x 1 4

8. If f (x) 5 4x 1 1 and g(x) 5 2x2, what is the value of g( f (28))?

1922 513 257 2127

2 4 6�2�4�6

2468

x

y

�4

O

A

G

C

H

A

G

A

F


page 61

page 63

page 62

page 64

Chapter 9 Project: Get the Picture

Beginning the Chapter ProjectWhen a book is being made, artists, designers, and photographers work with writers and editors to make the pages visually attractive. Th ese professionals often work with patterns involving arithmetic and geometric sequences.

In this project, you will see how perspective aff ects 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.

ActivitiesActivity 1: ResearchingResearch 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.

Activity 2: DesigningWhen a book is made, a designer or artist may change the size of an original sketch to fi t the space available on a page. One way to change the dimensions of a sketch is to use graph paper with diff erent size squares.

• Draw a fi gure or design on a sheet of graph paper. Label this Figure 1 and record its approximate dimensions.

• Enlarge the original fi gure 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 fi gure be? Draw this fi gure, label it Figure 3, and record its dimensions.

• Explain why the lengths of the three fi gures form a geometric sequence.• Write a geometric sequence corresponding to these lengths, and a recursive or explicit

formula for that sequence.

Figure 1 Figure 2

• 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.




83

A N S W E R S

Chapter 9 Project Manager: Get the Picture

Getting StartedRead 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 fi nd 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 Rubric4 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 satisfi es few of the requirements and shows poor organization and eff ort.

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


page 65 page 66

Chapter 9 Project: Get the Picture (continued)

Activity 3: AnalyzingPhotographs are often cropped so that only part of the photograph remains. Th en, 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 fi rst four terms of an arithmetic sequence that has the length of the diagonal of the cropped area as its fi rst term. Using the terms of your sequence as diagonal lengths, fi nd the four corresponding photo widths. What do you notice about this list of widths?

• Write the fi rst four terms of an geometric sequence that has the length of the diagonal of the cropped area as its fi rst term. Using the terms of your sequence as diagonal lengths, fi nd the four corresponding photo widths. What do you notice about this list of widths?

Finishing the ProjectTh e 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. Th en discuss the sequences you made.

Refl ect and ReviseReview 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 ProjectGeometric 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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