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ELL Support

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recursive formula

sequence

term

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

sequence

1. an ordered list of numbers

Mathematical Patterns

Geometry Suppose you are stacking boxes in levels that form squares. The numbers of boxes in successive levels form a sequence. The figure at the right shows the top four levels as viewed from above.

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

Think About a Plan

9-1

Mathematical Patterns

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?

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

1. How many boxes are in each of the first four levels?

explicit formula

1 z Level 1: z

term

3. each number in a sequence

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

4 z Level 2: z

9 z Level 3: z

16 z Level 4: z

2. How many boxes of equal size would you need for the next lower level? 25

recursive formula 3. What is a recursive or explicit formula that describes the number of boxes in the nth level? explicit: an 5 n2

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

initial condition

recursive formula

sequence

subscript number

term

4. How many boxes would you need to add three levels? 25 z z

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

sequence is the

initial condition

1

36 z z

1

49 z z

5

z 110 z

. term

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

5. What is a recursive or explicit formula that describes the total number of boxes in a stack of n levels? recursive: a1 5 1; an 5 an21 1 n2

in the

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

6. How can you use your formula to find the number of levels you will have

with a stack of 285 boxes?

a(n) subscript number .

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

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

explicit formula

9. An ordered list of numbers is called a(n) 10. The formula an11 5 an 1 5 is a(n)

.

an L 285

sequence

.

. 7. Suppose you are stacking a total of 285 boxes. Use your formula to find how

recursive formula .

many levels you will have. Show your work. a1 5 1, a2 5 1 1 22 5 5, a3 5 5 1 32 5 14, a4 5 14 1 42 5 30, a5 5 30 1 52 5 55, a6 5 55 1 62 5 91, a7 5 91 1 72 5 140, a8 5 140 1 82 5 204, a9 5 204 1 92 5 285

8. You need z 9 z levels to make a stack of 285 boxes.

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Practice

Form G

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Mathematical Patterns

Find the first six terms of each sequence. 1. an 5 22n 1 1 21, 23, 25, 27, 29, 211

2. an 5 n2 2 1 0, 3, 8, 15, 24, 35

3. an 5 2n2 1 1 3, 9, 19, 33, 51, 73

4. an 5 1n 1 1 2, 2, 2, 2, 2, 2

5. an 5 2n 1 2 4, 6, 10, 18, 34, 66

6. an 5 2n2 2 n 1, 6, 15, 28, 45, 66

7. an 5 4n 1 n2 5, 12, 21, 32, 45, 60

1 8. an 5 3 n3 64 125 1 8 3 , 3 , 9, 3 , 3 , 72

9. an 5 (22)n 22, 4, 28, 16, 232, 64

10. 214, 28, 22, 4, 10, c an 5 an21 1 6 where a1 5 214 13. 1, 3, 9, 27, c an 5 3an21 where a1 5 1 16. 36, 39, 42, 45, 48, c an 5 an21 1 3 where a1 5 36

11. 6, 5.7, 5.4, 5.1, 4.8, c 12. 1, 22, 4, 28, 16, c an 5 an21 2 0.3 where an 5 22an21 where a1 5 6 a1 5 1 1 1 1 1 2 1 2 14. 1, 2, 4, 8, 16, c 15. 3 , 1, 1 3 , 1 3, 2, c an 5 an21 1 13 where a1 5 23 an 5 12 an21 where a1 5 1 17. 36, 30, 24, 18, 12, c an 5 an21 2 6 where a1 5 36

18. 9.6, 4.8, 2.4, 1.2, 0.6, c an 5 12 an21 where a1 5 9.6

19. 7, 14, 21, 28, 35, c

20. 2, 8, 14, 20, 26, c

21. 5, 6, 7, 8, 9, c

23. 3, 5, 7, 9, 11, c an 5 2n 1 1; 41

24. 0.8, 1.6, 2.4, 3.2, 4, c an 5 0.8n; 16

5 1 1 3 25. 4, 2, 4, 1, 4, c

1 1 1 1 1 26. 2, 4, 6, 8, 10, c

2 2 2 2 2 27. 3, 13, 23, 33, 43, c

1 1 an 5 2n ; 40

an 5 n 2 13; 19 23

an 5 n4 ; 5

1 38. an 5 3 n explicit; 13, 23, 1, 43, 53

39. an 5 n2 2 6 explicit; 25, 22, 3, 10, 19

40. a1 5 5, an 5 3an21 2 7 recursive; 5, 8, 17, 44, 125

1 41. an 5 2 (n 2 1) explicit; 0, 12, 1, 1 12, 2

42. a1 5 5, an 5 3 2 an21 recursive; 5, 22, 5, 22, 5

43. a1 5 24, an 5 2an21 recursive; 24, 28, 216, 232, 264

46. The first figure of a fractal contains one segment. For each successive figure,

six segments replace each segment. a. How many segments are in each of the first four figures of the sequence? 1, 6, 36, 216 b. Write a recursive definition for the sequence. an 5 6an21 where a1 5 1 47. The sum of the measures of the exterior angles of any polygon is 3608. All the

angles have the same measure in a regular polygon. a. Find the measure of one exterior angle in a regular hexagon (six angles). 608 b. Write an explicit formula for the measure of one exterior angle in a regular polygon with n angles. an 5 360 n c. Why would this formula not be meaningful for n 5 1 or n 5 2? No polygon has one or two angles. 48. Reasoning In order to find a term in a sequence, its position in the sequence is doubled

and then two is added. What are the first ten terms in the sequence?

Find the eighth term of each sequence.

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

28. 1, 3, 5, 7, 9, c 15

29. 400, 200, 100, 50, 25, c 3.125

30. 0, 22, 24, 26, 28, c 214

31. 1, 2, 4, 8, 16, c 128

32. 44, 39, 34, 29, 24, c 9 1 1 35. 14, 2 2, 5, 10, 20, c 160

33. 0.7, 0.8, 0.9, 1.0, 1.1, c 1.4

34. 4, 11, 18, 25, 32, c c 53

Form G

Mathematical Patterns

45. Writing Explain how to find an explicit formula for a sequence. Look for a pattern in the sequence and ﬁnd a mathematical rule that gives the nth term, given the number n.

an 5 n 1 4; 24

22. 21, 0, 1, 2, 3, c an 5 n 2 2; 18

Practice (continued)

44. Error Analysis Your friend says the explicit formula for the sequence 1, 8, 27, 64 is an 5 n2. Is she correct? Explain. She is incorrect; in order to ﬁnd each term in the sequence, the term number must be cubed, not squared.

Write an explicit formula for each sequence. Find the twentieth term. an 5 6n 2 4; 116

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Determine whether each formula is explicit or recursive. Then find the first five terms of each sequence.

Write a recursive definition for each sequence.

an 5 7n; 140

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49. Writing Explain the difference between a recursive and an explicit formula. An explicit formula deﬁnes how to ﬁnd the nth term directly from the number n, while a recursive formula deﬁnes how to ﬁnd each term from the previous term(s). 50. Open-Ended Write five terms in a sequence. Describe the sequence using a recursive or explicit formula. Check students’ work.

36. 26, 29, 212, 215, 218, 227

37. A man swims 1.5 mi on Monday, 1.6 mi on Tuesday, 1.8 mi on Wednesday,

2.1 mi on Thursday, and 2.5 mi on Friday. If the pattern continues, how many miles will he swim on Saturday? 3.0 mi

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ANSWERS page 5

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page 6

Practice

9-1

Form K

Mathematical Patterns

Find the first five terms of each sequence. 1. an 5 4n 2 1

Form K

Mathematical Patterns

Write an explicit formula for each sequence. Then find the tenth term. 2. an 5 n2 1 4 5, 8, 13, 20, 29

Substitute 1 for n and simplify.

Practice (continued)

13. 7, 10, 13, 16, c

an 5 3n 1 4

14. 8, 9, 10, 11, 12, c an 5 n 1 7; 17

1 1 1 15. 2 2 , 0, 2 , 1, 1 2 , c

17. 3, 1, 21, 23, 25, c an 5 22n 1 5; 215

18. 1, 7, 25, 79, 241 an 5 3n 2 2; 59,047

an 5 12n 2 1; 4

a10 5 3(10) 1 4 5 z 34 z

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.

16. 1, 4, 9, 16, c an 5 n2; 100

The first five terms are 3, 7, z 11 z , z 15 z , and z 19 z .

1 3. a 5 2 n 1 2 2.5, 3, 3.5, 4, 4.5

4. an 5 3n

5. an 5 26n2 26, 224, 254, 296, 2150

3, 9, 27, 81, 243

19. Reasoning You and your friend are trying to find the 80th term in the

sequence 8, 14, 20, 26, 32, c. You use a recursive definition and your friend uses an explicit formula. Who will find the 80th term first? Why? Your friend will ﬁnd the 80th term ﬁrst because he is using an explicit formula. Your friend will substitute 80 into the formula to get the answer, while you will go through 79 iterations of the recursive formula.

6. Write an explicit formula for a sequence with 3, 5, 7, 9, and 11 as its first five terms. an 5 2n 1 1

20. Your neighbor recently began learning to play the guitar. On the first day, she

Write a recursive definition for each sequence. 7. 2, 6, 12, 20, c

practiced for 0.4 h. On the second day, she practiced for 0.5 h. She practiced for 0.65 h on the third day, and 0.85 h on the fourth day. If this pattern continues, how long will she practice on the seventh day? 1.75 h

8. 120, 60, 30, 15, c a1 5 120; an11 5 Q 12 R an

Identify the initial condition. a1 5 2 Use n to express the relationship between successive terms. an 5 an21 1 2n

21. Charles lost two rented movies, so he owes the rental store a fee of $40. At the

9. 3, 8, 13, 18, c a1 5 3; an11 5 an 1 5

10. 1, 3, 9, 27, c a1 5 1; an11 5 3an

end of each month, the amount that Charles owes will increase by 5%, plus a $2 billing fee. How much money will Charles owe the rental store after 8 months? $78.20

11. 2, 3, 8, 63, c a1 5 2; an11 5 an2 2 1

12. Writing Explain the difference between a recursive definition and an explicit

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

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Standardized Test Prep

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PED-HSM11A2TR-08-1103-009-L01.indd Sec1:6 3/25/09 7:22:58 PM

Mathematical Patterns

For Exercises 1−6, choose the correct letter. 1. What are the first five terms of the sequence? C

1. Consider the function f (x) 5 5x 1 1. Let the first term of a sequence be 0.

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

What is f (0)? Let f (0) be the second term of the sequence. Write the sequence. 0, 1 2. To create more terms of this sequence through iteration, continue to apply

f (x) to each output. The third term in this sequence can be described as f ( f (0)). What is the third term? f (f (0)) 5 f (1) 5 6

2. The formula an 5 3n 1 2 best represents which sequence? G

3, 6, 9, 12, 15

4, 7, 10, 13, 16

5, 8, 11, 14, 17

5, 9, 29, 83, 245

3. Determine the first 10 terms of this sequence. You already have the first 3 terms. 0; 1; 6; 31; 156; 781; 3906; 19,531; 97,656; 488,281 4. Determine the first 5 terms of the sequence formed through iterations of x f (x) 5 2 1 1. Begin with x 5 2. Describe the sequence. 2, 2, 2, 2, 2; all of the terms in the sequence are 2.

3. Which pattern can be represented by an 5 n2 2 3? D

4, 7, 12, 19, 28

1, 4, 9, 16, 25

22, 1, 6, 13, 22

5. Will you get the same type of sequence if you start with a different number? No; for example, if you start with x 5 0, the sequence is 0, 1, 1.5, c

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

an 5 4n

an 5

4n2

an 5

4n3

an 5

2n4

6. Iterations have uses other than to form numerical sequences. Consider this iterative

process, which forms a sequence of a set of three integers. Make a set of any three integers. Compute the absolute value of the difference between each pair of integers in the set. This produces a new set of three integers. Continue this process on each new set of three integers. Describe what eventually happens. No matter what three integers you

5. For the sequence 0, 6, 16, 30, 48, c, what is the 40th term? A

3198

3200

4000

16,000

choose to start with, the set will eventually repeat itself in combinations of the set {0, a, a}, where a is a positive integer. 7. You can form fractals through iterations. Fractals are geometric figures just like circles

6. A student sets up a savings plan to transfer money from his checking account

to his savings account. The first week $10 is transferred, the second week $12 is transferred, the third week $16 is transferred, and the fourth week $24 is transferred. If this pattern continues and he starts with $100 in his checking account, how many weeks will pass before his balance is zero? G 4

5

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Mathematical Patterns

You can define the terms in a sequence using an explicit formula or a recursive definition. You can use another method, called iteration, to form a sequence. The word iteration means to repeat an action. In mathematics, a sequence of numbers is generated through iteration when the same procedure is performed on each output.

Multiple Choice

21, 0, 5, 12, 21

Enrichment

6

or rectangles, but fractals have a special property that these geometric figures do not. You make fractals by iterating the figure itself. For example, start by drawing an equilateral triangle on graph paper. Divide each side into three equal parts. Draw another equilateral triangle on one side of the triangle that has the middle section as its base. Repeat this process on the remaining two sides. You have just created the first two iterations of a fractal called the Koch snowflake.

7

Short Response 7. After training for and running a marathon, an athlete wants to reduce her daily run

by half each day. The marathon is about 26 mi. How many days will it take after the marathon before she runs less than a mile a day? Show your work. [2] 5 days; Day 1: 13 mi, Day 2: 6.5 mi, Day 3: 3.25 mi, Day 4: 1.625 mi, Day 5: 0.8125 mi [1] correct answer, without work shown OR incorrect answer with correct sequence [0] incorrect answers and no work shown OR no answers given Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

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ANSWERS page 9

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Reteaching

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Reteaching (continued)

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Mathematical Patterns

Mathematical Patterns

To find a recursive definition for a sequence, you compare each term to the previous term.

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.

Problem

• If the terms become progressively larger, addition or multiplication may be involved.

What is the recursive definition for the sequence? 800, 2400, 200, 2100, 50, c To find the recursive definition for a sequence, first describe the sequence in words.

Problem

What is the next term in the sequence 6, 8, 11, 15, 20, c? The initial term is 800.

6

8 12

11 13

15 14

20 15

g and positive.

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

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

2400,

800,

200,

2100,

Find a pattern.

50, g

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

If the pattern is continued, the next term is 20 1 6, or 26.

Exercises

The next term in the sequence will be 225.

Now translate the description into the parts of the recursive formula.

Describe the pattern that is formed. Find the next three terms. 2. 3, 6, 12, 24, 48 3. 1, 22, 4, 28, 16, 232 Each term is increased by one more than the previous term; 20, 26, 33 Each term is multiplied by 2 to get the next term; 96, 192, 384 Each term is multiplied by 22 to get the next term; 64, 2128, 256

1. 1. 2. 3.

5, 6, 8, 11, 15

4. 4. 5. 6.

1, 3, 9, 27, 81

7. 7. 8. 9.

5, 25, 125, 625, 3125

10. 10. 11. 12.

The terms are alternatively negative g

Spread the numbers in the sequence apart, leaving space between numbers.

a1 5 800

The initial term is 800.

an 5 an21 4 (22)

To ﬁnd the next term, divide the previous term by 22.

Exercises Write a recursive definition for each sequence. 13. 38, 33, 28, 23, c an 5 an21 2 5 where a1 5 38

5. 100, 95, 90, 85, 80 6. 15, 18, 21, 24, 27 Each term is multiplied by 3 to get the next term; 243, 729, 2187 Each term is decreased by 5 to get the next term; 75, 70, 65 Each term is increased by 3 to get to the next term; 30, 33, 36

14. 7, 14, 28, 56, c an 5 2an21 where a1 5 7

16. 2, 6, 18, 54, c 17. 4.5, 5, 5.5, 6, c an 5 3an21 where a1 5 2 an 5 an21 1 0.5 where a1 5 4.5

8. 50, 49, 47, 44, 40 9. 240, 120, 60, 30, 15 Each term is multiplied by 5 to get the next term; 15,625; 78,125; 390,625 Each term is decreased by one more than the previous term; 35, 29, 22 Each term is divided by 2 to get the next term; 7.5, 3.75, 1.875

15. 25, 27, 29, 211, c an 5 an21 2 2 where a1 5 25 18. 17, 20, 24, 29, c an 5 an21 1 (n 1 1) where a1 5 17

3, 5, 9, 15, 23 11. 280, 120, 2180, 270, 2405 12. 1, 5, 13, 29, 61 In each term, the number is increased by two more than the previous term; 33, 45, 59 Each term is multiplied by 21.5 to get the next term; 607.5, 2911.25, 1366.875 Each term is multiplied by 2 and then 3 is added to get the next term; 125, 253, 509

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Arithmetic Sequences

9-2

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. a, a 1 d, a 1 2d, a 1 3d, c

Know

z

2, 5, 8, 11, 14, c

z

arithmetic

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

arithmetic

z

z

z

2. 8:15 a.m. is 495 min from 0 and 3:20 p.m. is 920 min from 0.

not arithmetic

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

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

z

1. If you define 12:00 a.m. as minute 0, then 6:43 a.m. is 403 min from 0.

Determine whether or not each sequence is arithmetic. 1. 1, 4, 7, 9, 11, c

Arithmetic Sequences

Transportation Suppose a trolley stops at a certain intersection every 14 min. The first trolley of the day gets to the stop at 6:43 a.m. How long do you have to wait for a trolley if you get to the stop at 8:15 a.m.? At 3:20 p.m.?

Arithmetic Sequence

Sample

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Think About a Plan

z

z

3. The trolley stops every 14 min .

not arithmetic

Need 4. To solve the problem I need to find:

Use the formula an 5 a 1 (n 2 1)d to find the indicated term in each arithmetic sequence. 5. Find the 12th term in the sequence that begins 3, 6, 9, c

36

6. Find the 38th term in the sequence that begins 4, 10, 16, c

226

7. Find the 104th term in the sequence that begins 5, 9, 13, c

417

the closest times that the trolley gets to the stop that are after 8:15 A.M. and 3:20 P.M.

.

Plan 5. What is an explicit formula for the number of minutes after 12:00 a.m. that the

trolley gets to the stop?

Arithmetic Mean

an 5 403 1 (n 2 1)14

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

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

x1y 2

Sample

The arithmetic mean of 4 and 6 is

416 10 2 5 2 5 5.

7. Using this n in your formula, when does the trolley stop? at 501 min How long do you have to wait for this trolley? 6 min

Find the missing number in the arithmetic sequence. This number is the arithmetic mean of the two given numbers. 8. c, 13,

, 37, c

25

9. c, 26,

, 42, c

34

10. c, 45,

, 99, c

72

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8. Use your formula to find the smallest n that gives the minutes just after 3:20 p.m. that the trolley arrives at the stop. 38

9. Using this n in your formula, when does the trolley stop? at 921 min How long do you have to wait for this trolley? 1 min

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ANSWERS page 13

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page 14

Practice

1. 2, 3, 5, 8, c no

2. 0, 23, 26, 29, c yes; 23

3. 0.9, 0.5, 0.1, 20.3, c yes; 20.4

4. 3, 8, 13, 18, . . . yes; 5

5. 14, 215, 244, 273, c yes; 229

6. 3.2, 3.5, 3.8, 4.1, c yes; 0.3

7. 234, 228, 222, 216, c yes; 6

8. 2.3, 2.5, 2.7, 2.9, c yes; 0.2 10. 11, 13, 17, 25, c no

35. an21 5 5, an11 5 11 8

36. an21 5 17, an11 5 3 10

37. an21 5 28, an11 5 29 28.5

38. an21 5 20.6, an11 5 3.8 1.6

39. an21 5 y 2 z, an11 5 y y 2 2z

40. an21 5 2t 1 3, an11 5 4t 2 1 3t 1 1

41. Open-Ended Write an arithmetic sequence of at least five terms with a positive common difference. a ﬁve-term sequence with a positive common difference

Find the 43rd term of each sequence. 11. 12, 14, 16, 18, c 96

12. 13.1, 3.1, 26.9, 216.9, c 2406.9

13. 19.5, 19.9, 20.3, 20.7, c 36.3

14. 27, 24, 21, 18, c 299

15. 2, 13, 24, 35, c 464

16. 21, 15, 9, 3, . . . 2231

17. 1.3, 1.4, 1.5, 1.6, c 5.5

18. 22.1, 22.3, 22.5, 22.7, c 210.5

19. 45, 48, 51, 54, c 171

20. 20.073, 20.081, 20.089, c 20.409

42. Error Analysis On your homework, you write that the missing term in the arithmetic sequence 31, ___, 41, c is 35 12 . Your friend says the missing term is 36. Who is correct? What mistake was made? Your friend is correct. You did not take the average of 31 and 41 correctly to ﬁnd the missing term of 36. 43. Reasoning Explain why 84 is the missing term in the sequence 89, 86.5, ___, 81.5, c. The common difference in the arithmetic sequence is 22.5, which means the missing term must be 84 as that is 2.5 less than the term before it and 2.5 more than the term after it. 44. Writing Describe the general process of finding a missing term in an arithmetic sequence. If the term that is missing occurs between two other terms that are consecutive to the missing term, you can take the arithmetic mean of the two terms. If the term that is missing is not consecutive, use the formula an 5 a 1 (n 2 1)d.

Find the missing term of each arithmetic sequence. 21. c 23, 7 , 49, c 36

22. 14, 7 , 28, c 21

24. c 14, 7 , 15, c 14.5

25. c 245, 7 , 239, c 242 26. c 25, 7 , 22, c −3.5

23. c 29, 7 , 33, c 31

27. 22, 7 , 2, c 0

28. c 26, 7 , 2, c 22

45. You are making an arrangement of cubes in concentric rings for a sculpture.

The number of cubes in each ring follows the pattern below. 1, 9, 17, 25, 33, c

29. 234, 7 , 77, c 21.5

30. c 245, 7 , 212, c 228.5 31. 22, 7 , 456, c 227

Form G

Arithmetic Sequences

Find the arithmetic mean an of the given terms.

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

9. 127, 140, 153, 166, c yes; 13

Practice (continued)

9-2

Form G

Arithmetic Sequences

a. Is this an arithmetic sequence? Explain. Yes; there is a common difference of 8. b. What are the next three terms? 41, 49, 57 c. If the sequence continues to the 100th term in this pattern, what will that term be? 793

32. c 34, 7 , 345, c 189.5

33. A teacher donates the same amount of money each year to help protect the

rainforest. At the end of the second year, she has donated enough money to protect 8 acres. At the end of the third year, she has donated enough money to protect 12 acres. How many acres will the teacher’s donations protect at the end of the tenth year? 40 acres

46. Each year, a volunteer organization expects to add 5 more people to the

number of shut-ins for whom the group provides home maintenance services. This year, the organization provides the service for 32 people. a. Write a recursive formula for the number of people the organization expects to serve each year. an 5 an21 1 5 where a1 5 32 b. Write the first five terms of the sequence. 32, 37, 42, 47, 52 c. Write an explicit formula for the number of people the organization expects to serve each year. an 5 32 1 5(n 2 1) d. How many people would the organization expect to serve in the 20th year? 127 people

34. Writing Explain how you know that the sequence 109, 105, 101, 97, 93, cis arithmetic. The sequence has a common difference between terms of 24.

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Prentice Hall Gold Algebra 2 • Teaching Resources

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13

14

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page 15 Practice

2. 6, 10, 14, 18, 22, c arithmetic; 4

42153

Form K

Arithmetic Sequences

Find the missing term of each arithmetic sequence.

Determine whether each sequence is arithmetic. If so, identify the common difference. 1. 1, 4, 7, 10, c

Practice (continued)

9-2

Form K

Arithmetic Sequences

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page 16

z

z

23 , 37, c 16. c9,

15. c4, ___, 18, c

3. 1, 3, 6, 10, 15, c not arithmetic

Find the arithmetic mean of the given terms. 4 1 18 5 22 22 4 2 5 11

72453

z

z

11 . The missing term is

10 2 7 5 3 This sequence is arithmetic. 3 z . The common difference is z 4. 216, 213, 29, 24, 2, c not arithmetic

z

z

17. 46, 37 , 28, c

5. 2, 9, 16, 23, 30, c arithmetic; 7

6. 43, 56, 69, 82, c arithmetic; 13

z

z

18. 212, 28 , 24, c

z

z

19. c4, 220 , 244, c

20. Error Analysis Your friend used the arithmetic mean to find the missing term

in the following sequence: 3, ___, 29, 42, c. His answer was 13. What error did your friend make? What is the correct answer?

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

arithmetic? Explain.

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

Yes; the difference between consecutive terms is 4. 21. An architect is designing a building with sides in the shape of a trapezoid. The

Find the 24th term of each arithmetic sequence. 8. 4, 6, 8, 10, 12, c

9. 2, 5, 8, 11, 14, c

an 5 a1 1 (n 2 1)d

an 5 a1 1 (n 2 1)d

a24 5 4 1 (24 2 1)2

71

number of windows on each floor forms an arithmetic sequence. There are 124 windows on the first floor and 116 windows on the second floor. a. Write an explicit formula to represent the sequence. an 5 132 2 8n b. How many windows are on the tenth floor? 52 windows

10. 9, 5, 1, 23, 27, c 283

a24 5 4 1 46 a24 5 z 50 z

22. Your cousin opened a bank account with a deposit of $256 dollars. After one

week, she had $280 in her account. After two weeks, she had $304, and after three weeks she had $328. If this pattern continues, how much money will your cousin have in her account after 18 weeks? $688

Find the missing terms in the following arithmetic sequences. 11. 2, ___, ___, 14, c

z

zz

z

9 , 15 12. 3, , 21, c

z

zz

z

13. 65, 54 , 43 , 32, c

14 5 2 1 3d 12 5 3d

23. There is a puddle 1.4 cm deep in your backyard. After one minute of rain, the

puddle was 1.45 cm deep. The puddle was 1.5 cm deep after it rained for two minutes. If the pattern continues, how deep will the puddle be after it rains for 45 min? 3.65 cm

d54 6 z 2 1 4 5 z

6 1 4 5 z 10 z 14. Error Analysis Noah used the formula an 5 a 1 (n 2 1)d to find the 12th

term in the sequence 2, 4, 7, 11, 16, c. Did Noah find the correct term? How do you know? No; Noah applied the explicit formula for arithmetic sequences to a

Prentice Hall Foundations Algebra 2 • Teaching Resources

sequence that is not arithmetic.

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ANSWERS page 17

page 18

Standardized Test Prep

9-2

For Exercises 1−6, choose the correct letter.

0

F1: e 1 , 11 f

1. Which sequence is an arithmetic sequence? A

7, 10, 13, 16, 19, c

7, 14, 28, 56, 112, c

7, 8, 10, 13, 17, c

1, 7, 14, 22, 31, 41, c

0

F2: e 1 , 12 , 11 f 0 1 1 2 1

F3: e 1 , 3 , 2 , 3 , 1 f

2. An arithmetic sequence begins 4, 9, c. What is the 20th term? I

80

84

0 1 1 1 2 3 1

F4: e 1 , 4 , 3 , 2 , 3 , 4 , 1 f

99 Each Farey sequence is a list of fractions in increasing order between 0 and 1, written in simplest form with a denominator less than or equal to the integer n. For any n greater than 1, there are an odd number of terms in the sequence and the middle term is 12 .

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

19, 24

19, 34

24, 43

43, 62

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

30

35

37

Problem

40

What are the terms of the Farey sequence for n 5 5? The Farey sequence for n 5 5 contains all the terms of the Farey sequence F4 plus the fractions between 0 and 1 which have a denominator of 5 when written in simplest form.

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

161

180

185

221

0

1 2 3

tiles in the second row, 18 green tiles in the third row, and 25 green tiles in the fourth row. If she continues the pattern, how many green tiles will she use in the 20th row? I 58

134

5

0

The fractions 5 and 5 will not be added because they simplify to 1 and 11 . Insert the 4

fractions 5 , 5 , 5 , and 5 in the Farey sequence F4.

6. An artist is creating a tile mosaic. She uses 4 green tiles in the first row, 11 green

32

Arithmetic Sequences

There are many types of sequences. One interesting type of sequence is the Farey sequence. The first four Farey sequences are:

Multiple Choice

76

Enrichment

9-2

Arithmetic Sequences

0

3

3

F5: e 1 , 15 , 14 , 13 , 25 , 12 , 5 , 23 , 4 , 45 , 11 f

Exercises

137

1. How many terms are in each of the first five Farey sequences? 2, 3, 5, 7, 11

Extended Response

2. What are the terms for the Farey sequence F6? e 01, 16, 15, 14, 13, 25, 12, 35, 23, 34, 45, 56, 11 f

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

3. What will be the new terms in the Farey sequence F7? e 17, 27, 37, 47, 57, and 67 f

Show your work. [4] 1587; a 5 3, n 5 100, d 5 16, an 5 a 1 (n 2 1)d; a100 5 3 1 (100 2 1)16 5 3 1 1584 5 1587 [3] appropriate method shown, with one computational error [2] appropriate method shown, with several computational errors OR correct term found incorrectly with work shown [1] incorrect term, without work shown [0] incorrect answers and no work shown OR no answers given

page 19

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Reteaching

4. Since 11 is a prime number, how many more terms will be in the sequence F11 compared to the sequence F10? 10 5. Is there any limit to how large n can be? No, n can be any positive integer although the computations become tedious. 6. Can you give examples of any other sequences? Answers may vary. Sample: arithmetic, geometric, and Fibonacci

• a is the starting value and d is the common difference.

Problem

• n is always greater than or equal to 1.

As a part-time home health care aide, you are paid a weekly salary plus a fixed fuel fee for every patient you visit. You receive $240 in a week that you visit 1 patient. You receive $250 in a week that you visit 2 patients. How much will you receive if you visit 12 patients in 1 week?

• You can write the sequence as a, a 1 d, a 1 2d, a 1 3d, c Problem

Find the 15th term of an arithmetic sequence whose first three terms are 20, 16.5, and 13.

a15 5 20 1 (15 2 1)(23.5)

d 5 a2 2 a1 5 250 2 240 5 10 a 5 240

First, ﬁnd the common difference. The difference between consecutive terms is 3.5. The sequence decreases. The common difference is 23.5.

n 5 12 an 5 a 1 (n 2 1)d

Use the explicit formula. Substitute a 5 20, n 5 15, and d 5 23.5.

5 20 1 (14)(23.5)

Subtract within parentheses.

5 20 1 249

Multiply.

5 229

The 15th term is 229.

Arithmetic Sequences

To solve word problems that involve arithmetic sequences, identify the common difference d, the starting value a, and the number of terms in the sequence n.

The explicit formula for the nth term of an arithmetic sequence is an 5 a 1 (n 2 1)d.

an 5 a 1 (n 2 1) d

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Reteaching (continued)

9-2

3/25/09 7:25:06 PM

Arithmetic Sequences

20 2 16.5 5 3.5 16.5 2 13 5 3.5

page 20

PED-HSM11A2TR-08-1103-009-L02.indd Sec2:18

The common difference is the difference between two consecutive terms. You receive $10 per visit. Identify the starting value. You receive $240 for a week with 1 visit. You want to ﬁnd the earnings in a week in which you visit 12 patients. 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

Check the answer. Write a1, a2, c, a15 down the left side of your paper. Start with a1 5 20. Subtract 3.5 and record 16.5 next to a2. Continue until you find a15.

7. Suppose you begin to work selling ads for a newspaper. You will be paid $50/wk

plus a minimum of $7.50 for each potential customer you contact. What is the least amount of money you earn after contacting eight businesses in 1 wk? $110

Exercises

8. A boy starts a savings account for a mountain bike. He initially deposits $15. He decides to increase each deposit by $8. How much is his 17th deposit? $143

Find the 25th term of each sequence. 1. 20, 18, 16, 14, c 228

2. 0.0057, 0.0060, 0.0063, c 0.0129

3. 4, 0, 24, 28, c 292

4. 0.2, 0.7, 1.2, 1.7, c 12.2

5. −10, 28.8, 27.6, 26.4, c 18.8

6. 22, 26, 30, 34, c 118

9. A woman is knitting a blanket for her infant niece. Each day, she knits four

more rows than the day before. She knitted seven rows on Sunday. How many rows will she knit on the following Saturday? 31 rows 10. Joe started a 30-min workout program this week. He wants to increase the

workout by 5 min every week. How long will his program be in the 16th week? 105 min

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page 22

ELL Support

Think About a Plan

9-3

Geometric Sequences

Use the chart below to review vocabulary. These vocabulary words will help you complete this page. Vocabulary Words

Explanations

Examples

Geometric sequence

A sequence in which the ratio of any term (after the first) to its preceding term is a constant value.

The sequence 3, 6, 12, 24, . . . is geometric because all of the consecutive terms have a ratio of 2.

Common ratio

the ratio of each term to its preceding term in a geometric sequence

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

Geometric mean

The geometric mean of two numbers x and y is !xy .

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

Geometric Sequences

Athletics During your first week of training for a marathon, you run a total of 10 miles. You increase the distance you run each week by twenty percent. How many miles do you run during your twelfth week of training?

Understanding the Problem

1. The terms in the sequence 2, 6, 18, 54, 162, . . . all share a

1. How can you write a sequence of numbers to represent this situation? Answers may vary. Sample: Start with 10, and multiply it and each successive term by 120% or 1.2

.

geometric

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

3. What is the first term of the sequence? 10

common ratio

with their preceding terms. 2. The numbers 8 and 2 have a

geometric mean

of 4. 4. What is the common ratio of the sequence? 1.2

3. The consecutive terms in a geometric sequence all share a common ratio.

Identify each sequence as arithmetic or geometric. 4. 2, 8, 32, 128, . . .

geometric

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

geometric

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

arithmetic

5. What is the problem asking you to determine? the 12th term of a geometric sequence that represents the number of miles you run each week

Identify the common ratio for each geometric sequence. 7. 3, 12, 48, 192, . . .

4

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

5

Planning the Solution 6. Write a formula for the sequence. an 5 10(1.2)n21

Find the missing term in the geometric sequence. 9. . . . , 4, ___, 16, . . .

8

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

15

Getting an Answer 7. Evaluate your formula to find the number of miles you run during your

twelfth week of training. about 74.3

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page 23

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Practice

9-3

Form G

Practice (continued)

Form G

PED-HSM11A2TR-08-1103-009-L03.indd 22

Geometric Sequences

Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms.

Determine whether each sequence is geometric. If so, find the common ratio. 1. 3, 9, 27, 81, . . . yes; 3

2. 4, 8, 16, 32, . . . yes; 2

4. 4, 28, 16, 232, . . . yes; 22

5. 1, 0.5, 0.25, 0.125, . . . yes; 0.5 6. 100, 30, 9, 2.7, . . . yes; 0.3

7. 25, 0, 5, 10, . . . no

3. 4, 8, 12, 16, . . . no

14. 23, 212, 248, . . . 2786,43215. 25, 25, 2125, . . . 9,765,625

1 1 1 1 16. 3, 9, 27 , . . . 59,049

17. 0.3, 0.6, 1.2, . . . 153.6

neither; 29, 214 arithmetic; 211, 214

38. a1 5 3, r 5 22 an 5 3(22)n21; 3, 26, 12, 224, 48

1 1 18. 4 , 2 , 1, . . . 128

41. a1 5 22, r 5 23 an 5 22(23)n21; 22, 6, 218, 54, 2162

19. When a pendulum swings freely, the length of its arc decreases geometrically.

44. a1 5 9, r 5 2 an 5 9(2)n21; 9, 18, 36, 72, 144

Find each missing arc length. a. 20th arc is 20 in.; 22nd arc is 18.5 in. about 19.2 in. b. 8th arc is 27 mm; 10th arc is 3 mm 9 mm c. 5th arc is 25 cm; 7th arc is 1 cm 5 cm d. 100th arc is 18 ft; 98th arc is 2 ft 6 ft

34. 2, 22, 2, 22, . . . geometric; 2, 21 37. 1, 22, 3, 24, . . . neither; 5, 26

39. a1 5 5, r 5 3 an 5 5(3)n21; 5, 15, 45, 135, 405

40. a1 5 21, r 5 4 an 5 21(4)n21; 21, 24, 216, 264, 2256 1 42. a1 5 32, r 5 20.5 43. a1 5 2187, r 5 3 1 an 5 32(20.5)n21; 32, 216, an 5 2187 Q 3 R n21 ; 2187, 8, 24, 2 729, 243, 81, 27 45. a1 5 24, r 5 4 46. a1 5 0.1, r 5 22 an 5 24(4)n21; 24, 216, an 5 0.1(22)n21; 0.1, 264, 2256, 21024 20.2, 0.4, 20.8, 1.6

47. The deer population in an area is increasing. This year, the population was

1.025 times last year’s population of 2537. a. Assuming that the population increases at the same rate for the next few years, write an explicit formula for the sequence. an 5 2537(1.025)n21 b. Find the expected deer population for the fourth year of the sequence. about 2732

Find the missing term of each geometric sequence. It could be the geometric mean or its opposite. 20. 4, j , 16, . . . 68

21. 9, j , 16, . . . 612

22. 2, j , 8, . . . 64

23. 3, j , 12, . . . 66

24. 2, j , 50, . . . 610

25. 4, j , 5.76, . . . 64.8

26. 625, j , 25, . . . 6125

1 27. 3 , j

28. 0.5, j , 0.125, . . . 60.25

, 3, . . . 61

36. 1, 22, 25, 28, . . .

Write an explicit formula for each sequence. Then generate the first five terms.

12. 22, 6, 218, . . . 39,366

11. 1, 3, 9, . . . 19,683

33. 1, 0, 22, 25, . . .

35. 23, 2, 7, 12, . . . arithmetic; 17, 22

Find the tenth term of each geometric sequence.

13. 23, 9, 227, . . . 59,049

1 32. 9, 3, 1, 3 , . . .

1 geometric; 19, 27

8. 64, 232, 16, 28, . . . yes; 20.59. 1, 4, 9, 16, . . . no

10. 2, 4, 8, . . . 1024

3/25/09 7:26:15 PM

Geometric Sequences

48. You enlarge the dimensions of a picture to 150% several times. After the first

increase, the picture is 1 in. wide. a. Write an explicit formula to model the width after each increase. an 5 1(1.5)n21 b. How wide is the photo after the 2nd increase? 1.5 in. c. How wide is the photo after the 3rd increase? 2.25 in. d. How wide is the photo after the 12th increase? about 86.5 in.

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

geometric.

Find the missing terms of each geometric sequence. (Hint: The geometric mean of positive first and fifth terms is the third term. Some terms might be negative.)

The sequence has a common ratio of 12 or 0.5 between terms. 30. Open-Ended Write a geometric sequence of at least seven terms. any seven-term sequence with a common ratio 31. Error Analysis A student says that the geometric sequence 30, __, 120 can be

49. 12, j , j , j , 0.75 6, 3, 1.5 or 26, 3, 21.5

completed with 90. Is she correct? Explain.

50. 29, j , j , j , 22304 236, 2144, 2576 or 36, 2144, 576

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

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

51. 6th term 1458

52. 19th term 2,324,522,934

53. nth term 6(3)n21

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ANSWERS page 25

9-3

page 26

Practice

9-3

Form K

Geometric Sequences

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

Form K

Geometric Sequences

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

1. 1, 3, 9, 27, c

2. 2, 5, 8, 11, 14, c not geometric Find the ratios between consecutive terms. 3 9 27 5 5 3 9 1

14. 2, z w12 z , 72, c

13. 5, ___, 45, c

Find the geometric mean of 5 and 45.

The sequence is geometric.

!xy

3 z. The common ratio is z 3. 22, 24, 28, 216, c geometric; 2

Practice (continued)

!45 ? 5 4. 500, 50, 5, 0.5, c 1 geometric; 10

!225

5. 0, 25, 50, 75, 100, c not geometric

z w15 z

1 6. Open-Ended Write a geometric sequence with a common ratio of 4. Explain

how you developed the sequence.

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

1 234 z , 2 1 , c 15. 4 , z 4

Find the ninth term of each geometric sequence. 7. 3, 12, 48, 192, c

8. 2, 6, 18, 54, c 13,122

Use the explicit formula. an 5 a1 ? rn21

missing term in the geometric sequence 4, ___, 256. Her answer was 130. What error did your classmate make? What is the correct answer?

a9 5 3(65,536)

z

17. 1.2, z w7.2 z , 43.2, c

18. Error Analysis On a recent math test, your classmate was asked to find the

a9 5 3(48) a9 5 196,608

16. 175, z w35 z , 7, c

9. 1875, 375, 75, 15, c 0.0048

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

z

Find the missing terms of each geometric sequence. 10. 2, ___, ___, 128, c

19. The bacteria population in a petri dish was 14 at the beginning of an

2 z , z 4 z , 8, c 11. 1, z

36 z , z 12 z , 4, c 12. 108, z

experiment. After 30 min, the population was 28, and after an hour the population was 56.

Identify the common ratio.

a. Write an explicit definition to represent this sequence. an 5 14 ? 2n21 b. If this pattern continues, what will be the bacteria population after 4 h? 3584

an 5 a1 ? rn21 a4 5 2r421 128 5 2r3 64 5 r3 45r

20. A corporation earned a profit of $420,000 in its first year of operation. Over the

8 z. The second term is z

next 10 years, the company’s CEO hopes to increase the profit by 8% each year. If the CEO reaches her goal, what will be the company’s profit in its seventh year, to the nearest dollar? $666,487

32 z . The third term is z

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Standardized Test Prep

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Geometric Sequences

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Enrichment Geometric Sequences

Multiple Choice

Doubling Periods in Geometric Sequences

For Exercises 1−6, choose the correct letter.

Consider the geometric sequence 3, 4 12, 6 4, 10 18, . . . .

3

1

1. Describe how the terms of the sequence are related. Each term is 12 times the preceding term.

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

40

180,224

262,144

2,883,584

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

The doubling period of a geometric sequence is the number of terms needed to reach a term at least twice as large as a given term. What is the doubling period for the given sequence? 2 terms

2. Which sequence is a geometric sequence? H

1, 3, 5, 7, 9, . . .

2, 4, 8, 16, 32, . . .

12, 9, 6, 3, 0, . . .

22, 26, 210, 214, 218, . . .

3. Write the first ten terms of the geometric sequence a1 5 3, r 5 1.1 to two decimal places. 3, 3.3, 3.63, 3.99, 4.39, 4.83, 5.31, 5.85, 6.43, 7.07

3. Which could be the missing term of the geometric sequence 5, __, 125, . . .? A

25

50

75

4. What is the doubling period for a1 5 3? for a2 5 3.3? 8 terms; 8 terms

100

Although the doubling period does not depend on which term is given, it does depend on the common ratio. For what value(s) of r is the doubling period of a geometric sequence greater than 1? 1 R | r | R 2

3 4. What could be the missing term of the geometric sequence 212, __,24 , . . .? H

24

26.375

3

4 The idea of a doubling period applies to certain everyday situations. For example, under optimum conditions, bacteria reproduce by splitting in two. Their numbers increase geometrically over time. Suppose at noon on a certain day, there are 1000 bacteria in a dish. At 6 p.m. on the same day, there are 8000 bacteria.

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

1

6

36

1,679,616

5. If a count is taken every hour, how many terms are in the geometric sequence? What is the common ratio? What is the doubling period? 7; !2; 2 terms

6. In each successive round of a backgammon tournament, the number of

6. If a count is taken every 40 min, how many terms are in the sequence? What is 3 the common ratio? What is the doubling period? 10; "2; 3 terms

players decreases by half. If the tournament starts with 32 players, which rule could predict the number of players in the nth round? I 32 5 (0.5)n

32 5 0.5r n21

an 5 15n21

an 5 (32)(0.5)n21

7. In both cases, how many hours does it take the bacteria to double? 2

Short Response 7. What is the 6th term of the geometric sequence 100, 50, . . .? Show your work

using the explicit formula. 1 [2] 3.125; an 5 ar n21 ; an 5 100 Q 1 R n21 ; a6 5 100 Q R 5 5 3.125; 2 2 correct term with work shown [1] incorrect term OR correct answer, without work shown [0] incorrect answers and no work shown OR no answers given

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page 30

Reteaching

9-3

• A geometric sequence has a constant ratio between consecutive terms. This number is called the common ratio.

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?

Problem

Step 1 Identify key information in the problem. You know that your friend’s rent was $1000 in 2000. This means a 5 1000. You also know that her rent in 2003 was $1092.73. This means that a4 5 1092.73. Her rent is raised by the same percent each year, which is the same as multiplying by a constant (e.g., a 5% increase is the same as multiplying by 1.05).

Find the 12th term of the geometric sequence 5, 15, 45, . . . . 5, 15, 45, . . . 15

45

an 5

Find r by calculating the common ratio between consecutive terms. This is a geometric sequence because there is a common ratio between consecutive terms.

5(3)n21

Step 2 Identify missing information. You need to find the common ratio r in order to find the rent in 2009, a10.

Substitute a 5 5 and r 5 3 into the explicit formula to ﬁnd a formula for the nth term of the sequence.

a12 5 5(3)11

Substitute n 5 12 to ﬁnd the 12th term of the sequence.

a12 5 885,735

Remember to ﬁrst calculate 311, then multiply by 5.

Step 3 Use the explicit formula to find r. an 5 arn 2 1

Exercises 1. 4, 2, 1, . . . Find a10.

Substitute a 5 1000, a4 5 1092.73, and n 5 4.

1092.73 5 1000r3

Simplify.

1.09273 5

Divide each side by 1000.

2 2 3. 6, 22, 3 , . . . Find a12. 259,049

5. 100, 200, 400, . . . Find a9. 25,600

6. 8, 32, 128, . . . Find a4. 512

1 7. a1 5 1, r 5 2 8. a1 5 2, r 5 3 9. 1 an 5 1 Q 12 R n21 ; 1, 12, 14, 18, 16 an 5 2(3)n 2 1; 2, 6, 18, 54, 162 1 1 10. a1 5 1, r 5 4 11. a1 5 5, r 5 10 12. 1 n21 1 1 1 1 1 1 an 5 5 Q 10 R ; 5, 12, 20 , 200 , 2000 an 5 1 Q 14 R n21; 1, 14, 16 , 64, 256 14. a1 5 1, r 5 3

Take the cube root of both sides.

Step 4 Use the value of r to find the rent in 2009, a10. arn 2 1

Write the explicit formula.

a10 5 (1000)(1.03)1021

Write the explicit formula for each sequence. Then generate the first five terms.

13. a1 5 5, r 5 2

r3

1.03 5 r

10,935 15 45 2. 5, 2 , 4 , . . . Find a8. 128

an 5

2 4 64 4. 1, 23 , 9 , . . . Find a7. 729

Write the explicit formula.

1092.73 5 (1000)r4 2 1

Find the indicated term of the geometric sequence. 1 128

Geometric Sequences

Problem

• A geometric sequence can be described by a recursive formula, an 5 an21 ? r , or as an explicit formula, an 5 a ? r n21 .

r 5 5 5 15 5 3

Reteaching (continued)

9-3

Geometric Sequences

a1 5 12, r 5 3

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.

an 5 12(3)n 2 1; 12, 36, 108, 324, 972

Exercises

a1 5 1, r 5 13

22. An athlete is training for a bicycle race. She increases the amount she bikes by

1 1 an 5 1 Q 13 R n21; 1, 13, 19, 27 , 81

the same percent each day. If she bikes 10 mi on the first day, and 12.1 mi on the third day, how much will she bike on the fifth day? By what percent does she increase the amount she bikes each day? 14.641 mi; 10%

15. a1 5 3, r 5 6

an 5 1(3)n 2 1; 1, 3, 9, 27, 81

an 5 3(6)n 2 1; 3, 18, 108, 648, 3888 1 16. a1 5 3, r 5 3 17. a1 5 2, r 5 2 18. a1 5 2, r 5 2 an 5 3(3)n 2 1; 3, 9, 27, 81, 243 an 5 2(2)n 2 1; 2, 4, 8, 16, 32 an 5 2 Q 12 R n21 ; 2, 1, 12, 14, 18

23. By clipping coupons and eating more meals at home, your family plans to decrease

1 1 19. a1 5 1, r 5 5 20. a1 5 3, r 5 4 21. a1 5 5, r 5 4 5 5 5 1 1 1 an 5 3(4)n 2 1; 3, 12, 48, 192, 768 an 5 5 Q 14 R n21 ; 5, 54, 16 , 64, 256 , 125 , 625 an 5 1 Q 15 R n21 ; 1, 15, 25

24. From 2005 to 2009, a teen raised her babysitting rates by a fixed percent every

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

their monthly food budget by the same percent each month. If they budgeted $600 in January and $514.43 in April, how much will they budget in December? $341.28 year. If she charged $8/h in 2005 and $10.04/h in 2007, how much did she charge in 2009? What is her percent of increase each year? $12.59/h; 12%

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page 31 ELL Support

9-4

Arithmetic Series

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

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Think About a Plan Arithmetic Series

Architecture In a 20-row theater, the number of seats in a row increases by three with each successive row. The first row has 18 seats.

Column B

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

1. arithmetic series

A. a series that continues without end

b. Find the total seating capacity of the theater.

2. finite series

B. the sum of the terms of a sequence

c. Front-row tickets for a concert cost $60. After every 5 rows, the ticket price goes down

3. infinite series

C. a series whose terms form an arithmetic sequence

4. limits

D. the least and greatest values of n in a series

5. series

E. a series with a first and a last term

by $5. What is the total amount of money generated by a full house? 1. Write the explicit formula for an arithmetic sequence. an 5 a1 1 (n 2 1)d 2. What are a1 and d for the sequence that represents the number of seats

Identify the following series as finite or infinite. 6. 2 1 6 1 18 1 54

in each row?

ﬁnite

7. 3 1 10 1 17 1 24 1 c

inﬁnite

8. 2 1 10 1 50 1 250 1 c

inﬁnite

a1 5 z 18 z

d 5 z 3 z

3. Write an explicit formula for the arithmetic sequence that represents the

number of seats in each row. an 5 3n 1 15

9. Circle the arithmetic series in the group below.

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

1 1 4 1 7 1 10 1 13

4 1 14 1 24 1 34 1 44

20

4 1 6 1 10 1 12 1 16

a (3n 1 15) n51

2 1 4 1 8 1 16 1 32

0 1 12 1 24 1 36 1 48

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

1 1 6 1 36 1 216 1 1296

Answers may vary. Sample: Use the sum command and the sequence command 10. Complete the summation notation for the following sequence by filling in the

upper and lower limits.

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

u

3 1 11 1 19 1 27 1 35 1 c 1 115 5 a (8n 2 5) lower limit: n 5 1; upper limit: 15

.

6. Find the total seating capacity of the theater. 930

u

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

10

15

20

n51

n56

n511

n516

a (3n 1 15); a (3n 1 15); a (3n 1 15); a (3n 1 15)

8. Use your graphing calculator to evaluate each series.

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

120; 195; 270; 345

$60; $55; $50; $45

10. What is the total amount of money generated by a full house? $46,950

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ANSWERS page 33

9-4

page 34

Practice

9-4

Form G

Arithmetic Series

1. 1 1 3 1 5 1 7 1 9 25

2. 5 1 8 1 11 1 c 1 26 124

3. 4 1 9 1 14 1 c 1 44 216

4. (210) 1 (225) 1 (240) 1 c 1 (285) 2285

5. 17 1 25 1 33 1 c 1 65 287

6. 125 1 126 1 127 1 c 1 131 896

n51

11. 3 1 7 1 11 1 c 1 31

a (2n 2 1)

n51

sequence. A series is the sum of terms in a sequence, which is indicated by summation notation or addition signs. 38. a. Open-Ended Write three explicit formulas for arithmetic sequences. b. Write the first seven terms of each related series. c. Use summation notation to rewrite the series. d. Evaluate each series. Check students’ work. Sequences should be arithmetic and contain seven terms. 39. Error Analysis A student identifies the series 10 1 15 1 20 1 25 1 30 as an

13. 15 1 25 1 35 1 c 1 75 7

10

a (10n 1 5)

a (25n 2 15)

n51

n51

Find the sum of each finite series. 6

16. a (n 1 25) 183

n52 4

10

1

6

19. a (3 2 n) 23

n51

n51

4

41. To train new employees, an employer offers a bonus after 30 work days as

follows. An employee must turn in one report on the first day; the number of reports for each subsequent day must increase by two. What is the minimum number of reports an employee will have to turn in over the 30 days to earn the bonus? 900

6

21. a (2n 2 3) 222

n55

No; the series is a ﬁnite arithmetic series. An inﬁnite arithmetic series would continue indeﬁnitely. 3 40. Mental Math Use mental math to evaluate a (2n 1 1). 15

n53

18. a (2n 1 0.5) 22

20. a n 45

infinite arithmetic series. Is he correct? Explain.

8

15. a (2n 2 1) 35

5

34. 10 1 20 1 30 1 40 1 50 series; ﬁnite

37. Writing Explain how you can identify the difference between a series and a

8

a (4n 2 1)

n51

12. (220) 1 (225) 1 (230) 1 c 1 (265)

n52

33. 40, 20, 10, 5, 2.5, 1.25, c sequence; inﬁnite

the second has 11, the third has 13 and so on. How many musicians are in the last row and how many musicians are there in all? 19 musicians; 84 musicians

n51

7

17. a (5n 1 3) 82

32. 1 1 5 1 9 1 c 1 21 series; ﬁnite

6

9. 10 1 7 1 4 1 c 1 (25) a (23n 1 13)

8. 4 1 8 1 12 1 16 a 4n

4

31. 8, 8.2, 8.4, 8.6, 8.8, 9.0, c sequence; inﬁnite

36. A marching band formation consists of 6 rows. The first row has 9 musicians,

Write each arithmetic series in summation notation.

n51

30. 3 1 5 1 7 1 9 1 c series; inﬁnite

stitches in each successive row. The 25th row, which is the last row, has 77 stitches. Find the total number of stitches in the pattern. 1025 stitches

shelf below it. The bottom three shelves are 36 in., 31 in., and 26 in. wide. a. The shelf widths decrease by the same amount from bottom to top. What is the width of the top shelf? 6 in. b. What is the total shelf space of all seven shelves? 147 in.

4

29. 7, 12, 17, 22, 27 sequence; ﬁnite

35. An embroidery pattern calls for five stitches in the first row and for three more

7. A bookshelf has 7 shelves of different widths. Each shelf is narrower than the

14. a (n 2 1) 6

Form G

Arithmetic Series

Determine whether each list is a sequence or a series and finite or infinite.

Find the sum of each finite arithmetic series.

10. 1 1 3 1 5 1 c 1 13

Practice (continued)

22. a (3n 1 2) 62

n51

n53

Use a graphing calculator to find the sum of each series. 15

12

23. a (n 1 3) 165

20

24. a (2n 2 1) 144

n51

25. a 2n2 5740

n51

25

26. a (n3 1 2n) 106,275 n51

n51

50

27. a (n2 2 4n) 37,825 n51

25

28. a (5n3 1 3n) 528,570 n55

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PED-HSM11A2TR-08-1103-009-L04.indd Sec1:34

33

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9-4

9-4

Form K

Arithmetic Series

an 5 a1 1 (n 2 1)d 99 5 1 1 (n 2 1)2 99 5 1 1 2n 2 2 100 5 2n

270 5 5n

an 5 3 1 (n 2 1)5

54 5 n

Write the summation notation.

u 54

z

z

22 R a Q 5n

n51

15. 5 1 7 1 9 1 c 1 131

58

Find the number of terms. Find the sum.

n Sn 5 2 (a1 1 an) an 5 a1 1 (n 2 1)d 50 S50 5 2 (1 1 99) 406

Find the value of n for 268. 268 5 5n 2 2

an 5 a1 1 (n 2 1)d

14. 1 1 7 1 13 1 c 1 343

8. 3 1 7 1 11 1 15 1 c 1 55

Find the number of terms. Find the sum.

Find an explicit formula for the nth term.

an 5 5n 2 2

Find the sum of each finite arithmetic series. 7. 1 1 3 1 5 1 c 1 99

Form K

Arithmetic Series

13. 3 1 8 1 13 1 c 1 268

3. 4, 10, 16, 22, 28 ﬁnite sequence

5. 1.4 1 1.1 1 0.8 1 0.5 1 . . . 6. 22 2 11 2 20 2 29 2 c inﬁnite series inﬁnite series

4. 5 1 12 1 19 1 26 1 33 ﬁnite series

Practice (continued)

Write each arithmetic series in summation notation.

Identify each list as a series or a sequence and finite or infinite. 2. 1 1 4 1 7 1 10 1 13 ﬁnite series

page 36

3/25/09 8:26:49 PM

Practice

1. 2, 6, 10, 14, c inﬁnite sequence

3/25/09 8:26:51 PM

n Sn 5 2 (a1 1 an)

64

a (6n 2 5)

a (2n 1 3)

n51

n51

16. Tabitha used tiles to make the design shown at the right. The first

column has 2 tiles, the second column has 4 tiles, and the pattern continues. a. Write an explicit formula for the sequence. an 5 2n b. Write the summation notation for a related series with 24 tiles 12 a (2n) in the 12th column. n51 c. How many tiles are in the design if there are a total of 12 columns?

5 25(100)

5 z 2500 z

50 5 n

156 tiles

9. 106 1 101 1 96 1 c 1 1 1177

10. 2 1 10 1 18 1 c 1 378 9120

17. Your brother is preparing for basketball season. He shot 26 baskets on the first

day that he practiced. He shot 32 baskets on the second day and 38 baskets the day after that. a. If this pattern continues, how many baskets will he shoot on the 30th day? 200 baskets b. How many baskets will he have shot during those 30 days? 3390 baskets

11. (24) 1 (29) 1 (214) 1 c 1 (299) 21030

12. Reasoning Is it possible to find the sum of an infinite arithmetic series? Explain. No; an inﬁnite arithmetic series has a never-ending number of terms, so it is impossible to add them all.

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36

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ANSWERS page 37

9-4

page 38

Standardized Test Prep

Enrichment

9-4

Arithmetic Series

Arithmetic Series

Multiple Choice

The Gauss Trick

For Exercises 1–6, choose the correct letter.

If your teacher asked you to add the numbers from 1 to 100, you would probably begin by adding 1 1 2 1 3 1 c1 100, term by term from left to right. Karl Friedrich Gauss (1777– 1855) found another way. Let S represent the finite series whose sum you are trying to find. Since addition is commutative, both equations below represent this series. S 5 001 1 92 1 93 1 c1 98 1 99 1 100

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

2450

2500

2550

4950

S 5 100 1 99 1 98 1 c1 93 1 92 1 001

2. Which of the following is an infinite series? H

3, 8, 13, 18, 23

3 1 8 1 13 1 18 1 23 1 c

3 1 8 1 13 1 18 1 23

3, 8, 13, 18, 23, c

1. What is the sum of the left side of the first equation and the left side of the second equation? 2S 2. What is the sum of each vertically-aligned pair of quantities on the right side of the equal signs? 101

3. The high school choir is participating in a fundraising sales contest. The choir

3. How may such pairs are there? 100

will receive a bonus if they make 20 sales in their first week and improve their sales by 3 in every subsequent week. What is the minimum number of sales the choir could make in the first 12 weeks to qualify for the bonus? C 13

53

438

4. Because each pair has the same sum, use multiplication to express the sum of all the pairs on the right side. 100 3 101 5 10,100

5015

5. Write an equation that states that the sum of the left sides must equal the sum of the right sides. Solve your equation for S. 2S 5 10,100; S 5 5050

4. What is summation notation for the series 5 1 7 1 9 1 c 1 105? F 51

51

a (2n 1 3)

50

a (n 1 3)

n51

Use the technique outlined above to derive the formula for the sum of n terms of any arithmetic series. Suppose that the series starts with the term a1 and has a common difference of d.

51

a (2n 1 3)

n51

a (n 1 3)

n51

n57

6. What is the nth term, in terms of a1, d, and n? a1 1 (n 2 1)d

100

5. What is the upper limit of the summation a (n 2 2)? D

7. Write the sum S of the n terms of the series, where each number is written in

n51

1

2

98

100

990

1980

terms of a1 and d. Then write the sum in reverse order, lining up terms.

S 5 a1 1 (a1 1 d ) 1 c1 fa1 1 (n 2 1)d g; S 5 fa1 1 (n 2 1)d g 1 c1 (a1 1 d ) 1 a1 8. What is the sum of each vertical pair of quantities on the right side? 2a1 1 (n 2 1)d 9. How many such pairs are there? n

30

6. What is the sum of the series a (2n 1 2)? H n51

62

66

10. Express the sum of all the pairs using multiplication. nf2a1 1 (n 2 1)dg 11. Write an equation that states that the sum of the left sides must equal the sum of

the right sides. Solve your equation for S. 2S 5 nf2a1 1 (n 2 1)dg; S 5 nf2a1

Short Response 7. What is the sum of the finite arithmetic series 2 1 4 1 6 1 c 1 50? Show your work.

1 (n 2 1)dg 2

n

12. Show that your equation is equivalent to S 5 2 (a1 1 an). Hint: Use your answer to Exercise 6. nf2a1 1 (n 2 1)dg S5 2

[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

5 5 5

nfa1 1 a1 1 (n 2 1)dg 2 nfa1 1 fa1 1 (n 2 1)dgg 2 nfa1 1 an g 5 n2 (a1 1 an) 2

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9-4

PED-HSM11A2TR-08-1103-009-L04.indd Sec2:37

Reteaching

3/25/09 8:26:54 PM

Arithmetic Series

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

3/25/09 8:26:57 PM

Reteaching (continued)

9-4

Arithmetic Series

Problem

The debate club is offering a prize at the end of 10 weeks to a current member who brings three new members for the first meeting, and then increases the number of new members they bring each week by two thereafter. One member qualified for the prize with the minimum number of new members. How many new members did the member bring at Week 10? For all 10 weeks?

To find the sum of an arithmetic series written in summation notation: n

• list the terms and add them, or use the formula Sn 5 2 (a1 1 an) Problem

What is the sum of the series written in summation notation? Step 1 Identify key information in the problem. To win the prize, a member must bring three members to the first meeting, so a 5 3. A member must also bring two more new members to each meeting, so d 5 2. The contest extends for 10 weeks, so n 5 10.

4

a. a (5 2 2n) n52

4

a (5 2 2n)

Circle the upper and lower limits. Box the explicit formula.

n52

n52

n53

n54

n52

n53

n54

5 2 2(2)

5 2 2(3)

5 2 2(4)

4

a (5 2 2n) 5 5 2 2(2) 1 5 2 2(3) 1 5 2 2(4) 1

1 23

(21)

1

Step 2 Identify the information you are trying to find. You want to find the 10th term, a10 , and the sum of the first 10 terms, S10 .

Under each circle copy the explicit formula, substituting the value in the circle for n.

Step 3 Use the explicit formula to find a10 . an 5 a 1 (n 2 1)d a10 5 3 1 (10 2 1)2 a10 5 21

The value of the series is the sum of the values in the boxes.

n52

5 5

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

(23)

Evaluate each expression.

To win the prize, a member brought 21 new members to a meeting at Week 10.

Find the sum of the terms.

The sum of the series is 23.

Step 4

Use the value of a10 to find the total number of new members brought by the winner.

15

Sn 5 n2 (a1 1 an)

n51

S10 5 2 (3 1 21) S10 5 120

b. a (4n 2 1)

n Use the formula Sn 5 2 (a1 1 an).

a1 5 4(1) 2 1 5 3 an 5 a15 5 4(15) 2 1 5 59 15

Sn 5 2 (3 1 59) 5 465

Write the explicit formula. Substitute a 5 3, d 5 2, and n 5 10. Simplify.

10

First, ﬁnd n, a1 , and an . The upper limit is 15. Evaluate the explicit formula at n 5 1.

Write the formula for the sum of an arithmetic series. Substitute a1 5 3, a10 5 21, and n 5 10. Simplify.

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

Evaluate the explicit formula at n 5 15. Substitute n 5 15, a1 5 3, and an 5 59.

Exercises

Simplify.

9. The seating arrangement for a recital uses 20 seats in the first row and two

The sum of the series is 465.

additional seats in each row thereafter. How many seats will be in the eighth row? In the ninth row? How many seats total are there in the first nine rows?

Exercises

34 seats; 36 seats; 252 seats

Find the sum of each finite series. 3

1. a (n 2 4) 26 n51 9

5. a (4 2 2n) 256 n53

4

1 10 2. a 3 n 3 n51 5

6. a 8n 120 n51

8

3. a (3n 2 1) 93 n53 7

7. a 4n 108 n52

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

8

2n 4. a 3 22

the first four quizzes: 65, 70, 75, and 80. If the scores continue to increase at this rate, what will be the score in the 7th week? In the 8th week? What is the total of the first eight scores? 95; 100; 660

n53 7

8. a (3 2 2n) 235 n51

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ANSWERS page 41

page 42

9-5

ELL Support

9-5

Geometric Series

Geometric Series

Communications Many companies use a telephone chain to notify employees of a closing due to bad weather. Suppose a company’s CEO calls three people. Then each of these people calls three others, and so on.

Problem

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

a. Make a diagram to show the first three stages in the telephone chain. How

6 18 54 2 5 6 5 18 5 3 nth term 5 1458

Identify the common ratio and the nth term.

many calls are made at each stage? b. Write the series that represents the total number of calls made through the

an 5 a1rn21

first six stages.

Use the explicit formula.

1458 5 2 ? 3n21

c. How many employees have been notified after stage six?

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 Sn 5

Think About a Plan

tree diagram

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

Use a calculator. 2. Make a diagram to show the first three stages in the telephone chain.

a1(1 2 rn) 12r

Use the sum formula.

2(1 2 37) S7 5 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? 16 64 4 1 5 4 5 16 5 4

Identify the common ratio and the nth term .

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

nth term 5 1024 an 5 a1rn21

4. Write the series that represents the total number of calls made through the first six stages. 3 1 9 1 27 1 81 1 243 1 729

Use the explicit formula

.

1024 5 1 ? 4n21

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

.

1024 5 4n21

Divide each side by 1

.

Use a calculator

.

n 6. Write the sum formula. S 5 a1(1 2 r ) n 12r

Use the sum formula

.

7. Use the sum formula to find how many employees have been notified

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

.

5. What is the sum of this series? 1092

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

a1(1 2 12r

rn)

1(1 2 46) S6 5 1 2 4

n

after stage six. S 5 a1(1 2 r ) 5 3(1 2 36) 5 1092 n 12r 123 8. Does your answer agree with your sum from Exercise 5? yes

S6 5 1365

PED-HSM11A2TR-08-1103-009-L05.indd 42

page 43 Practice

9-5

PED-HSM11A2TR-08-1103-009-L05.indd 41

Form G

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Geometric Series

2. 4 1 12 1 36 1 c; n 5 15 28,697,812

3. 15 1 12 1 9.6 1 c; n 5 40 about 74.99

4. 27 1 9 1 3 1 c; n 5 100 about 40.5

arithmetic; 126

7. This month, your friend deposits $400 to save for a vacation. She plans to

deposit 10% more each successive month for the next 11 months. How much will she have saved after the 12 deposits? $8553.71 Determine whether each infinite geometric series diverges or converges. State whether each series has a sum. 3

3

9. 4 1 2 1 1 1 c converges; yes

11. 6 1 11.4 1 21.66 1 c diverges; no

12. 220 2 8 2 3.2 2 c converges; yes

26. 4 1 8 1 16 1 32 1 c; n 5 15 geometric; 131,068

27. 5 1 10 1 15 1 20 1 c; n 5 20 arithmetic; 1050

`

2

15

13. 50 1 70 1 98 1 c diverges; no

17. 1000 1 750 1 562.5 1 421.875 1 c 4000

25. 22 1 2 1 6 1 10 1 c; n 5 12 arithmetic; 240

n51

`

29. a (22. 1)n21 n51

no sum

`

1

30. a Q 22 R n21 n51

2 3

`

5

31. a 2 Q 3 R n21 n51

no sum

32. Open Ended Write an infinite geometric series that converges to 2. Show your work. Check students’ work.

Evaluate each infinite geometric series.

16. 120 1 96 1 76.8 1 61.44 1 c 600

255 geometric; 1024

24. 23 1 6 2 12 1 24 2 c; n 5 10 geometric; 1023

28. a 5 Q 3 R n21

converges; yes

14. 8 1 4 1 2 1 1 1 c 16

1 1 1 1 23. 8 1 16 1 32 1 64 1 c; n 5 8

Evaluate each infinite series that has a sum.

10. 17 1 15.3 1 13.77 1 c

1 15. 1 1 13 1 19 1 27 1

Form G

Geometric Series

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

5. 0.2 1 0.02 1 0.002 1 c; n 5 8 0.22222222 6. 100 1 200 1 400 1 c; n 5 6 6300

8. 3 1 2 1 4 1 c converges; yes

Practice (continued)

Determine whether each series is arithmetic or geometric. Then evaluate the series for the specified number of terms.

Evaluate each finite series for the specified number of terms. 1. 40 1 20 1 10 1 c; n 5 10 79.921875

9-5

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page 44

. . . 1.5 Find the specified value for each infinite geometric series.

18. Suppose your business made a profit of $5500 the first year. If the profit increased 20% per year, find the total profit over the first 5 yr. $40,928.80 19. The end of a pendulum travels 50 cm on its first swing. Each swing after the

first, it travels 99% as far as the preceding swing. How far will the pendulum travel before it stops? 5000 cm

25 33. a1 5 5, S 5 3 , find r 2 5

1 34. S 5 108, r 5 3 , find a1 72

35. a1 5 3, S 5 12, find r 0.75

36. S 5 840, r 5 0.5, find a1 420

37. Error Analysis Your friend says that an infinite geometric series cannot have

a sum because it’s infinite. You say that it is possible for an infinite geometric series to have a sum. Who is correct? Explain.

20. A seashell has chambers that are each 0.82 times the length of the enclosing

chamber. The outer chamber is 32 mm around. Find the total length of the shell’s spiraled chambers. about 177.78 mm

You are; an inﬁnite geometric series with »r… less than 1 has a series of partial sums that converges towards a number. 38. Writing Describe in general terms how you would find the sum of a finite geometric series. Identify the ﬁrst term, common ratio, and nth term. Use the explicit formula to ﬁnd n. Then, use the sum formula with the ﬁrst term, common ratio, and n to ﬁnd the sum of the series.

21. The first year a toy manufacturer introduces a new toy, its sales total $495,000.

The company expects its sales to drop 10% each succeeding year. Find the total expected sales in the first 6 years. Find the total expected sales if the company offers the toy for sale for as long as anyone buys it. $2,319,367.05; $4,950,000

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ANSWERS page 45

page 46

Practice

9-5

9-5

Form K

Geometric Series

Find the number of terms. Use the sum formula. an 5 a1

4374 5 2 ? 3n21 2187 5

2. 1 1 2 1 4 1 c 1 2048 4095

1 1 8. 1 1 4 1 16 1 c

Because ur u 5 P 14 P , 1, the series converges. a 1 1 113 S 5 1 21 r 5 5 3 5 1 2 14 4

n Sn 5 a1(1 2 r ) 12r 2(1 2 38) S8 5 1 2 3

rn21

z

5 6560

3n21

Form K

Geometric Series

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

Find the sum of each finite geometric series. 1. 2 1 6 1 18 1 c 1 4374

Practice (continued)

z

9. 2 1 8 1 32 1 c diverges

z

z

37 5 2187

1 1 1 10. 2 1 16 1 128 1 c

n58

converges; 47

1 3. 8 1 4 1 2 1 c 1 256 N 16

3 9 1 11. 4 1 8 1 16 1 c diverges

2 2 12. 2 2 5 1 25 2 c converges; 123

13. Your classmate is trying to cut down on the amount of time he spends watching

4. 3 1 9 1 27 1 . . . 1 6561 9840

television. In January, he spent a total of 3600 min watching television. He watched television for 3240 min in February and 2916 min in March. If this pattern continues, how many minutes of television will he watch this year? about 25,833 min

5. 24 2 8 2 16 2 c 2 2048 24092 14. Your math teacher asks you to choose between two offers. The first offer is to

receive one penny on the first day, 3 pennies on the second day, 9 pennies on the third day, and so on, for 14 days. The second offer is to receive 4 pennies on the first day, 8 pennies on the second day, 16 pennies on the third day, and so on, for 14 days. Which offer is better? What is the difference between the total amounts received? the ﬁrst offer; 2,325,952 pennies

6. Find the sum of the geometric series 2 2 4 1 8 2 16 1 c 1 8192. Explain how you found the sum. 5462; I used the explicit formula to determine that there are 13 terms in the series. Then I used the sum formula to determine the sum.

7. A family farm produced 2400 ears of corn in its first year. For each of the next

9 yr, the farm increased its yearly corn production by 15%. How many ears of corn did the farm produce during this 10-yr period? 48,729

page 47 Standardized Test Prep

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9-5

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9-5

Geometric Series

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Enrichment Geometric Series

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

Gridded Response Solve each exercise and enter your answer in the grid provided. 20 1 n 1. What is the value of a1 in the series a 3 Q 2 R ?

1 1 1. Evaluate the expression 1 2 x for x 5 2 and for x 5 2 . 21; 2

n50

2. What is the sum of the geometric series 2 1 6 1 18 1 c1 486?

2. You can evaluate many expressions with a power series called a Taylor series. 1 To evaluate 1 2 x , you use the Taylor series 1 1 x 1`x2 1 x3 1 c. What is n this infinite series written in summation notation? a x

3. A community organizes a phone tree in order to alert each family of

3. Determine the sum of the first five terms of the Taylor series 1 1 1 x 1 x2 1 x3 1 cfor x 5 2 and for x 5 2 . 31; 1.9375

n50

emergencies. In the first stage, one person calls five families. In the second stage, each of the five families calls another five families, and so on. How many stages need to be reached before 600 families or more are called?

4. For which value of x is your computation above a better approximation for the 1 value of the expression 12x ? What might need to be true about the value of x in order for this Taylor series to converge to the value of this expression? 12;»x… R 1 5. You can use a different Taylor series to evaluate ex . Write the Taylor series ` n x x x2 x3 1 1 1! 1 2! 1 3! 1 cin summation notation. a n!

4. What is the approximate whole number sum for the finite geometric series 5

n50

n

1 a 8Q4 R ?

1 6. Evaluate ex for x 5 2 . Evaluate the first four terms of the Taylor series x x2 x3 1 1 1 1! 1 2! 1 3! 1 cfor x 5 2 . Round your answers to the nearest thousandth. 1.649; 1.646

n50

7. Does the Taylor series for ex still converge if u x u $ 1? Does it give you the

1 1 5. What is the sum of the geometric series 1 1 3 1 9 1 c? Enter your answer

same value as the function y 5 ex? Explain your reasoning.

as a fraction.

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.

Answers 1.

–

3 0 1 2 3 4 5 6 7 8 9

2. 0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

–

7 28 0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

3. 0 1 2 3 4 5 6 7 8 9

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

–

4 0 1 2 3 4 5 6 7 8 9

4. 0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

–

1 1 0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

5. 0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

–

3 / 2 0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

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ANSWERS page 49

page 50

Reteaching

9-5

a (1 2 rn)

• The sum of a finite geometric series is Sn 5 1 1 2 r , where a1 is the first term, r is the common ratio, and n is the number of terms.

Your neighbor hosts a family reunion every year. In 2000, it costs $1500 to host the reunion. Their expenses have decreased by 10% per year by asking family members to contribute food and party supplies. a. What is a rule for the cost of the family reunion? b. What was the cost of the reunion in 2005? c. What was the total cost for hosting the family reunions from 2000 to 2009?

a

Problem

What is the sum of the first ten terms of the geometric series 8 1 16 1 32 1 64 1 128 1 c?

16

32

64

128

an 5 arn21 an 5 (1500)(0.90)n21

Simplify the ratio formed by any two consecutive terms to ﬁnd r.

n 5 10 8(1 2 210) S10 5 1 2 2

5

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

a1 is the ﬁrst term in the series.

r 5 8 5 16 5 32 5 64 5 2

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

5 8184

Write the explicit formula. Substitute a1 5 1500, r 5 1 2 0.10 5 0.90 in the explicit formula.

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

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

8(21023) 21

Geometric Series

Problem

• The sum of an infinite geometric series with u r u , 1 is S 5 1 21 r , where a1 is the first term and r is the common ratio. If u r u $ 1, then the series has no sum.

a1 5 8

Reteaching (continued)

9-5

Geometric Series

an 5 arn21

Write the explicit formula.

Simplify inside the parentheses.

an 5 (1500)(0.90)621

Simplify.

an < 886 Simplify. The cost of hosting the reunion in 2005 was $886.

Exercises

Substitute a1 5 1500, r 5 0.90, n 5 6 in the formula.

10

To find the total of hosting the reunions from 2000 to 2009, a (1500)(0.90)n21 , n51 find the sum of the geometric series.

Evaluate the finite series for the specified number of terms. 1. 3 1 12 1 48 1 192 1 c; n 5 6 4095

1 1 2. 8 1 2 1 2 1 8 1 c; n 5 5 341 32

635 3. 210 2 5 2 2.5 2 1.25 2 c; n 5 7 2 32

5 5 3415 4. 10 1 (25) 1 2 1 Q 24 R 1 c; n 5 11 512

Sn 5 S10 5

Evaluate each infinite geometric series. 5. 10 1 5 1 2.5 1 c 20

7 49 2 4 1 11 6. 21 1 11 2 121 1 c 213 7. 4 1 32 1 256 1 c 2

1 1 2 5 8. 2 2 5 1 25 2 c 14

1 1 1 1 9. 26 1 12 2 24 1 c 29

4 11. 12 1 4 1 3 1 c 18

1 1 1 1 12. 4 2 8 1 16 2 c 6

a1(1 2 rn) 12r

1500(1 2 0.9010) 1 2 0.90

S10 < 9770

Write the formula for the sum of a geometric series. Substitute a1 5 1500, r 5 0.90, n 5 10 in the formula. Simplify.

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

64 10. 20 1 16 1 5 1 c 100

Exercise

2 2 2 5 13. 3 1 15 1 75 1 c 6

14. In 1990, a vacation package cost $400. The cost has increased 10% per year. a. What are the values of a1 and r? a1 5 400, r 5 1.1 b. What is a rule for the cost of the vacation? an 5 (400)(1.10)n21 c. What was cost of the vacation in 1995? $644.20 d. What was the total cost of the vacations from 1990 to 1999? $6374.97 e. If the pattern continued until 2009, what was the total cost of the vacations? $22,910 Prentice Hall Algebra 2 • Teaching Resources Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

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Chapter 9 Quiz 1

Chapter 9 Quiz 2

Form G

Lessons 9-1 through 9-2

Lessons 9-3 through 9-5

Do you know HOW?

Do you know HOW?

1. an 5

1 2n 3, 8, 15, 24, 35

1. 4, 8, 16, c 512

2. an 5 n 2 6 25, 24, 23, 22, 21

5

2. 20,480; 5120; 1280; c 4

Find the seventh term of each sequence.

Find the seventh term of each sequence. 3. 10, 9, 7, 4, c 211

3. 2, 4, 8, 14, c 44

4. 2, 4, 8, 16, c 128

5. 13, 19, 25, 31, c yes; d 5 6

4. 1, 23, 9, 227, c 729

Determine whether each sequence is arithmetic or geometric. Then evaluate the finite series for the specified number of terms.

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

5. 1 1 3 1 9 1 c; n 5 8 geometric; 3280

6. 16, 24, 36, 54, c no

Find the missing term of each arithmetic sequence. 7. 4, ___ , 24, 34, c 14

Form G

Find the eighth term of each geometric sequence.

Find the first five terms of each sequence.

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page 52

page 51

6. 25 1 32 1 39 1 c; n 5 12 arithmetic; 762

Evaluate each infinite geometric series. `

8. 100, ___, 92, c 96

1 n21 2 7. a Q 22 R 3 n51

`

8. a 3(0.4)n21 5 n51

Do you UNDERSTAND? 9. Vocabulary Explain what it means for a formula to be an explicit formula.

Do you UNDERSTAND?

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

9. Open-Ended Write an arithmetic series that has a negative sum. Answers may vary. Sample: 1.2 1 0.2 2 0.8 2 . . . 2 8.8 5 241.8

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

10. Reasoning Can an infinite geometric series converge when the common

Check students’ work.

ratio is greater than 1? Explain. Give an example. Answers may vary. Sample: No, the terms of the series grow in absolute value so they cannot have a ﬁnite sum; two possible series with r 5 2 are 3 1 6 1 12 1 24 1 . . . and (21) 1 (22) 1 (24) 1 . . . .

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ANSWERS page 53

page 54

Chapter 9 Chapter Test

Chapter 9 Chapter Test (continued)

Form G

Form G

Evaluate each infinite geometric series.

Do you know HOW?

22. 30 1 22.5 1 16.875 1 c 120

Write a recursive definition and an explicit formula for each sequence. Then find a10. 1. 41, 46, 51, 61, c 2. 1, 10, 100, 1000, 10000, c an 5 an21 1 5 where an 5 10an21 where a1 5 41; an 5 41 1 5(n 2 1) a1 5 1; an 5 1(10)n21; or an 5 36 1 5n; 86 1,000,000,000

23. 15 2 3 1 0.6 2 0.12 1 c 12.5 1 25. 4 2 2 1 1 2 2 1 c 83 5 5 27. 25 2 2 2 4 2 c 210

24. 12 1 6 1 3 1 c 24 1 1 26. 2132 1 9 2 6 1 c 28 10

3. 3, 6, 12, 24, 48, c an 5 2an21 where a1 5 3; an 5 3(2)n21; 1536

Determine whether each series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms.

Find the first five terms in each arithmetic sequence. 4. an 5 3n 1 2 5, 8, 11, 14, 17

5. an 5 n 1 5 6, 7, 8, 9, 10

28. 5 1 9 1 13 1 17 1 c; n 5 10 arithmetic; 230

29. 35 1 70 1 140 1 280 1 c; n 5 7 geometric; 4445

6. an 5 12n 12, 24, 36, 48, 60

7. an 5 2n 1 10 9, 8, 7, 6, 5

30. 6 1 (218) 1 54 1 (2162) 1 c; n 5 8 geometric; 29840

31. 8 1 11 1 14 1 17 1 c; n 5 6 arithmetic; 93

32. 10 1 8 1 6 1 4 1 c; n 5 10 arithmetic; 10

4 4 33. 20 1 4 1 5 1 25 1 c; n 5 6

Determine whether each sequence is arithmetic, geometric, or neither. Then find the ninth term. 8. 3, 12, 48, 192, c geometric; 196,608

On October 3, she plants 26 bulbs. She continues in this pattern until October 15, when she plants the last bulbs. a. Write an explicit formula to model the number of bulbs she plants each day. an 5 20 1 3(n 2 1) b. Write a recursive definition to model the number of bulbs she plants each day. an 5 an21 1 3 where a1 5 20 c. How many bulbs will the gardener plant on October 15? 62 bulbs d. What is the total number of bulbs she plants from October 1 to October 15, inclusive? 615 bulbs

9. 22, 27, 212, 217, c arithmetic; 242

7 1 11. 2, 2, 2, 5, c arithmetic; 25 2

2 2 2 10. 10, 2, 5, 25, c geometric; 78,125

Find the missing term of each geometric sequence. It could be its geometric mean or its opposite. 12. 16, 7, 4 6 8

13. 25, 7, 225 6 75

14. 2, 7, 50 6 10

15. 1, 7, 49 6 7

3 16. 4 , 7, 3 6 32

17. 36, 7, 4 6 12

15,624

geometric; 625

34. On October 1, a gardener plants 20 bulbs. On October 2, she plants 23 bulbs.

35. Suppose you are building 10 steps with 6 concrete blocks in the top step and 60 blocks

in the bottom step. If the number of blocks in each step forms an arithmetic sequence, find the total number of concrete blocks needed to build the steps. 330 blocks

Do you UNDERSTAND? 36. Writing Explain why an infinite geometric series with r 5 1 diverges. Include

an example in your explanation. `

Answers may vary. Sample: a 5(1)n21 5 5 1 5 1 5 1 . . .;The series diverges n51

Find the sum of each finite series.

because the number 5 is added an inﬁnite number of times

5

37. Open-Ended Write a sequence and describe it using both an explicit definition

8

18. a (n 2 1) 10

and a recursive formula.

19. a 3n 9840

n51

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

n51

15

20

20. a (3n 1 1) 375

38. Reasoning What does a recursive definition have that an explicit formula does not? Explain. Answers may vary. Sample: A recursive deﬁnition contains an initial condition as well as a formula for how to move from one term to the other. An explicit formula describes the nth term in terms of n.

21. a (5 2 n) 2110

n51

n51

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page 55

Chapter 9 Quiz 1

page 56 Chapter 9 Quiz 2

Form K

Lessons 9–1 through 9–2

Lessons 9–3 through 9–5

Do you know HOW?

Do you know HOW? Find the eighth term of each geometric sequence.

Find the first four terms of each sequence. 1. an 5 5n 2 2 3, 8, 13, 18

Form K

1. 4, 12, 36, 108, c 8748

1 3. an 5 22 n 2 12, 21, 232, 22

2. an 5 n3 1 5 6, 13, 32, 69

1 1 2. 2, 1, 2 , 4 , c N 0.0156

3. 0.04, 0.2, 1, 5, c 3125

Find the sum of each finite arithmetic series.

Write a recursive definition for each sequence. 4. 80, 40, 20, 10, c a1 5 80; an11 5 12an

5. 4, 10, 16, 22, c a1 5 4; an11 5 an 1 6

4. 3 1 6 1 9 1 c 1 72 900

6. 3, 21, 147, 1029, c a1 5 3; an11 5 7an

6. (22) 1 (27) 1 (212) 1 c 1 (2102) 21092

Find the 18th term of each arithmetic sequence. 7. 5, 9, 13, 17, c 73

8. 4, 1, 22, 25, c 247

9. 1.2, 1.6, 2, 2.4, c 8

Write each arithmetic series in summation notation. 7. 1 1 5 1 9 1 c 1 85 22

Use the arithmetic mean to find the missing term in each arithmetic sequence.

z

z

10. c6, 17 , 28, c

z

z

5. 6 1 12 1 18 1 c 1 108 1026

11. c2, 26 , 214, c

z

z

12. c1.4, 4.1 , 6.8, c

8. 5 1 11 1 17 1 c 1 371 62

a (4n 2 3)

a (6n 2 1)

n51

n51

9. 212 1 204 1 196 1 c 1 (220) 30

a (220 2 8n) n51

Do you UNDERSTAND?

Find the sum of each finite geometric series. 10. 5 1 15 1 45 1 c 1 10,935

13. Writing Describe the difference between a recursive definition and an explicit definition of a sequence. A recursive deﬁnition relates each term to the next. An explicit deﬁnition describes the nth term of a sequence using the number n.

16,400

1 1 1 1 11. 6 1 12 1 24 1 c 1 384 about 0.33

12. 1 2 4 1 16 2 c 2 16,384 213,107

14. Tim takes the stairs up to his office. He enters the ground floor of the building

and climbs 12 steps to reach the first floor. He climbs a total of 24 steps to reach the second floor and 36 steps to reach the third floor. How many steps will Tim climb to reach his office on the 16th floor? 192 steps

Do you UNDERSTAND? 13. A guitar-making company produced 60 guitars this month. The company

plans to increase production by 8% each month for the next 9 months. How many guitars will they produce during this 10-month period? about 869 guitars

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ANSWERS page 57

page 58

Chapter 9 Test

Chapter 9 Test (continued)

Form K

Do you know HOW?

Form K

Do you know HOW?

Find the first five terms of each sequence. 1. an 5 3n 1 4 7, 10, 13, 16, 19

Find the 9th term of each geometric sequence. 1 3. an 5 2 n 2 2 −1.5, −1, −0.5, 0, 0.5

2. an 5 n2 1 n 2, 6, 12, 20, 30

13. 5, 10, 20, 40, c 1280

Write an explicit formula for each sequence. Then find the 12th term. 4. 2, 6, 12, 20, c an 5 n(n 1 1); 156

5. 2.5, 3, 3.5, 4, c an 5 12 n 1 2; 8

8. 56, 50, 44, 38, c −58

15. 22, 210, 250, 2250, c −781,250

Find the sum of each finite arithmetic series. 16. 8 1 12 1 16 1 c 1 116 1736

6. 2, 5, 10, 17, 26, c an 5 n2 1 1; 145

17. (23) 1 (29) 1 (215) 1 c 1 (2201) −3468

18. 1 1 5 1 9 1 c 1 157 3160

Find the 20th term of each arithmetic sequence. 7. 2, 5, 8, 11, c 59

14. 32, 28, 2, 20.5, . . . N 0.0005

9. 2.2, 2.6, 3, 3.4, c 9.8

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

Do you UNDERSTAND?

1 19. 4, 2, 1, 2 , . . . converges; 8

10. Writing Find the missing term in the sequence below. Then explain how you

found the term.

20. 6, 18, 54, 162 diverges

21. 5, 21, 0.2, 20.04, . . . converges; 4.16

c12, z 28 z , 44, c

Do you UNDERSTAND?

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

22. Error Analysis Your friend calculated the sum of the finite geometric series

2 1 8 1 32 1 c 1 32,768. Her answer was 131,080. What error did she make? What is the correct sum?

11. Reasoning Rita must find the 35th term in the sequence that begins

2, 9, 16, 23, c. She needs to find the answer as fast as possible. Should Rita use a recursive definition or an explicit formula? Why? explicit formula; using a

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

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

23. Writing Find the possible values of the missing term in the following

geometric sequence, and explain how you found the answer.

12. A bus has 6 people on it as it pulls out of the station to begin its route. After

6, z ±24 z , 96, c

one stop, there are 11 people on the bus. After the second stop, there are 16 people on the bus. If this pattern continues, how many people will be on the bus after 10 stops? 51 people

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.

page 60

page 59 Chapter 9 Performance Tasks

Chapter 9 Performance Tasks (continued)

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Task 1 a. Use your graphing calculator to graph the function f(x) 5 2x over the domain 5x | x $ 06. b. Use the TABLE feature on your calculator to make a table of values of the function f for the set of x-values 1, 2, 3, . . . . c. Determine whether the sequence of function values is arithmetic, geometric, or neither. Justify your response. geometric; There is a common ratio of 2. d. Write a recursive definition and an explicit formula for the sequence of function values. an 5 2an21 where a1 5 2; an 5 2n e. Find three terms of the sequence between 512 and 8192, and identify these as arithmetic or geometric means. Explain your reasoning.

Task 3 a. Determine whether the sequence 2, 8, 14, 20, 26,cis arithmetic geometric, or neither. Justify your response. Arithmetic; there is a common difference of 6. an 5 6 1 an21 where b. Write a recursive definition and an explicit formula for this sequence. a1 5 2; an 5 2 1 6(n 2 1) c. Find three terms of the sequence between 62 and 86, and identify 68, 74, and 80; arithmetic these as arithmetic or geometric means. Explain your means; 74 is the average of 62 reasoning. and 86, 68 is the average of 62 and 74, and 80 is the average of 74 and 86. ` d. Use summation notation to write the series related to the infinite sequence a (6n 2 4); 290 given in part (a). Find the sum of the first ten terms of the series. n51 e. Describe a real-world situation that can be modeled by the sequence given in part (a). Check students’ work.

Y1 5 2ˆX

X55 X 1 2 3 4 5 6 X50

Y 5 32 Y1 2 4 8 16 32 64

[4] Student uses a calculator to view the function and constructs a table of values using positive integers for x-values. Student correctly identiﬁes the sequence as geometric and justiﬁes answer. Student writes a recursive deﬁnition and an explicit formula for the sequence, ﬁnds three terms, and identiﬁes these as arithmetic means with justiﬁcation. Student uses summation notation to write a series and correctly ﬁnds the sum of the ﬁrst ten terms. Student describes a real-world situation. [3] Student completes all parts with minor errors. [2] Student makes major errors in one or more parts. [1] Student determines minimal and/or incorrect information about the sequence, its formulas, and the geometric means. There are major errors in logic. [0] Student makes no attempt, or no response is given.

1024, 2048, and 4096; geometric means; 2048 is the square root of the product of 512 and 8192, 1024 is the square root of the product of 512 and 2048, and 4096 is the square root of the product of 2048 and 8192. [4] Student correctly uses calculator to view the function and constructs table of values using positive integers for x-values. Student correctly identiﬁes the sequence as geometric and justiﬁes answer. Student correctly writes recursive deﬁnition and explicit formula for the sequence, ﬁnds three terms, and identiﬁes these as geometric means with justiﬁcation. [3] Student correctly completes parts (a), (b), and (c). Justiﬁcation may not be fully developed. Student completes parts (d) and (e) with only minor errors and some justiﬁcation. [2] Student correctly completes parts (a), (b), and (c). Justiﬁcation is not given. Student writes recursive deﬁnition and explicit formula with one or more errors. Student ﬁnds three terms with one or more major errors. Justiﬁcation is not given. [1] Student determines minimal and/or incorrect information about the sequence, its formulas, and the geometric means. There are major errors in logic. [0] Student makes no attempt, or no response is given.

Task 4 a. Graph the function f (x) 5 20.5x2 1 4.5 for the domain 23 # x # 3 using your graphing calculator. b. Carefully draw the graph of the function on a sheet of graph paper. Check student’s drawing. c. Draw and use inscribed rectangles 1 unit wide to approximate the area 23 under the curve for the given interval. 13 units2 d. Use e f (x)dx feature from the CALC menu of your graphing calculator to determine the area under the curve for the given interval. 18 units2

Task 2 a. Determine whether the sequence 27, 9, 3, 1, . . . is geometric, arithmetic, or neither. Justify your response. geometric; There is a common ratio of 13 . an 5 13 an21 where b. Write a recursive definition and an explicit formula for this sequence. a1 5 27; an 5 27 Q 13 R n21 10 c. Use summation notation to write the series related to the first ten terms 1 n21 29,524 ; 729 a 27 Q 3 R of the sequence give in part (a). Then evaluate this series. n51 d. Use summation notation to write the series related to the infinite ` 81 1 n21 27 Q R ; converges; sequence given in part (a). Determine whether this series diverges or a 3 2 n51 coverages. If the series converages, find its sum. e. Describe a real-world situation that can be modeled by this sequence.

10

3

25

[4] Student correctly graphs the function over the designated domain on a graphing calculator. Student draws a neat graph of the function on graph paper. Student makes a close estimate of the area under the curve using rectangles. Student correctly uses a graphing calculator to ﬁnd the area under the curve. [3] Student completes all parts with only minor errors. [2] Student makes major errors in one or more parts. [1] Student determines minimal and/or incorrect information about the graph and the area under it. There are major errors in logic. [0] Student makes no attempt, or no response is given.

[4] Student correctly determines that the sequence is geometric. Student correctly ﬁnds a recursive deﬁnition and explicit formula for the sequence. Student correctly writes the series using summation notation, ﬁnds the sum of the ﬁrst ten terms of the series, determines that the inﬁnite series converges, and correctly determines the sum. Student describes a feasible real-world situation. [3] Student completes all parts with only minor errors. Student describes a feasible real-world situation. [2] Student completes all parts with one or more major errors. [1] Student determines minimal and/or incorrect information about the sequence and series, their recursive deﬁnitions and explicit formulas, and the series in summation notation. There are major errors in logic. [0] Student makes no attempt, or no response is given.

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ANSWERS page 61

page 62

Chapter 9 Cumulative Review

Chapter 9 Cumulative Review (continued)

Multiple Choice

Short Response

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

(0, 0.75)

(0, 2.25)

(3, 0)

1 8x 1 15 in simplest form? G x2 2 x 2 12 x15 x15 x24 x13

2. What is

9. Graph the system of inequalities e

(4, 0)

"x23 2

3 ! x

quadratic

The graph of y 5 log3(x 2 2) 1 5 is a shift of the graph of the parent function y 5 log3x to the right two units and up ﬁve units.

3 2 " x

cubic

x 2 4 6 8

11. How can the relationship between variables in the table be described? The variables in the table, x and y, have an inverse variation x y relationship. As x-values increase, y-values decrease, but the product of their pairs remains constant. 1 20

4. How is the polynomial 2x2 2 x3 1 4x 1 17 classified by degree? H

linear

y

⫺4

graph of the parent function.

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

x15 x14

3 "x2

2 !x

⫺4 ⫺2 O

10. Describe how the graph of y 5 log 3(x 2 2) 1 5 compares to the

x2

3. Which expression is equivalent to 6 ? C "x2

8 6 4 2

y , 2x 2 1 . y $ 2x 1 3

quartic

2

10

4

5

5

4

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

There is one real solution.

There is one complex solution.

There are no real solutions.

There are two complex solutions.

12. Use the sequence 100, 95, 90, 85, . . . This is an arithmetic sequence in which each term is a. Describe the sequence in words. ﬁve less than the previous one. b. Find the next three terms. 80, 75, 70

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

log28

log39

log464

log5125

7. Which inequality is graphed? A

y#x14

y$x14

y#x24

y,x14

8 6 4 2 ⫺6 ⫺4 ⫺2 O

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

1922

513

13. Water leaks from a 10,000-gal tank at a rate of 5 gal/h. Write a linear model for the situation and use it to find the amount of water in the tank after 24 h. w 5 25t 1 10,000; 9880 gal

y

Extended Response x

14. You have a coupon for $10 off a CD. You also get a 20% discount if you show

2 4 6

your membership card in the CD club. How much more would you pay if the cashier applies the coupon first? Use composite functions. Show your work.

⫺4

257

[4] $2; student deﬁnes both functions and subtracts one from the other correctly. [3] Student deﬁnes both functions and subtracts one from the other with minor errors. [2] Student determines minimal and/or incorrect information about the functions and does not subtract one from the other. There are major errors in logic. [1] Student provides incorrect information. No work is shown. [0] Student makes no attempt or no response is given.

2127

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Chapter 9 Project Teacher Notes: Get the Picture

page 64 Chapter 9 Project: Get the Picture

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About the Project

Beginning the Chapter Project

The Chapter Project gives students an opportunity to use sequences, explicit formulas, and recursive formulas to change the sizes of drawings and photos. They investigate perspective, the use of grids to enlarge and reduce, and ways to crop, enlarge, and reduce photographs.

When a book is being made, artists, designers, and photographers work with writers and editors to make the pages visually attractive. These professionals often work with patterns involving arithmetic and geometric sequences. In this project, you will see how perspective affects perceived lengths and distances. You will use grids to change the sizes of drawings. You also will learn how a designer crops a photo, then enlarges or reduces it.

Introducing the Project • Ask students if they have ever seen artists draw buildings or other objects that appear to recede in the distance.

Activities Activity 1: Researching

• Ask them why it appears that railroad track rails get closer together when we look at them in the distance.

Research the concepts of one- and two-point perspective and vanishing points in art. • Measure the lengths of the arrows shown at the right. What is the relationship between these lengths? How does this relate to your research on perspective?

• Explain that they will investigate the concepts of perspective and vanishing points, and the mathematics involved in enlarging, reducing, and cropping pictures and photographs.

• 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?

Activity 1: Researching Students research perspective, create drawings in perspective, write arithmetic sequences, and determine explicit or recursive formulas for their sequences.

• 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: Designing Students use grid paper to enlarge designs. They use the same ratios repeatedly to draw lengths which form geometric sequences. They then write explicit or recursive formulas for their sequences.

• Check students’ work; answers may vary. Sample: The lengths form an arithmetic sequence; answers may vary. Sample: The lines of sight along the tops and bottoms of the arrows meet at a vanishing point. • Check students’ work; answers may vary. Sample: There is no longer a vanishing point. • Check students’ work.

Activity 3: Analyzing

Activity 2: Designing

Students crop photos. Then they enlarge the cropped portions, writing sequences for the widths of the enlargements.

Figure 1

Figure 2

When a book is made, a designer or artist may change the size of an original sketch to fit the space available on a page. One way to change the dimensions of a sketch is to use graph paper with different size squares. • Draw a figure or design on a sheet of graph paper. Label this Figure 1 and record its approximate dimensions. • Enlarge the original figure by copying each portion of Figure 1, square by square, onto larger squares. Label this Figure 2 and record its dimensions. • Use a ratio to compare the dimensions of Figure 1 to the dimensions of Figure 2. If the same ratio is used to enlarge Figure 2, what would the dimensions of the new figure be? Draw this figure, label it Figure 3, and record its dimensions. • Explain why the lengths of the three figures form a geometric sequence. • Write a geometric sequence corresponding to these lengths, and a recursive or explicit formula for that sequence. Check students’ work.

Finishing the Project You may wish to plan a project day on which students share their completed projects. Encourage students to explain their processes as well as their results. • Have students review their methods for writing explicit and recursive formulas for arithmetic and geometric sequences. • Ask groups to share their insights that resulted from completing the project, such as any shortcuts they found for making their drawings or writing formulas.

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ANSWERS page 65

page 66

Chapter 9 Project: Get the Picture (continued)

Chapter 9 Project Manager: Get the Picture

Activity 3: Analyzing

Getting Started

Photographs are often cropped so that only part of the photograph remains. Then, this cropped portion can be reduced or enlarged. Choose a photograph in a textbook. Place a piece of paper over the photograph, trace its original size, and draw a rectangle to indicate a portion of the photograph that you would like to crop. Draw a diagonal from the lower left corner to the upper right corner of the rectangular cropped area. If this diagonal is extended through the upper right corner of the cropped area, and a point selected anywhere along the diagonal or its extension, then the rectangle having the chosen point as its upper right corner (and the same lower left corner as the original cropped area) will have dimensions that are proportional to the dimensions of the cropped area. • Measure the dimensions and the length of the diagonal of the cropped area.

Read the project. As you work on the project, you will need a calculator, a metric ruler, at least two types of graph paper, and materials on which you can record your calculations. Keep all of your work for the project in a folder.

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

Checklist

Suggestions

☐ Activity 1: relating perspective and arithmetic sequences

☐ Use art books from the school library or the Internet.

☐ Activity 2: relating dimensions and geometric sequences

☐ Use grid paper to draw simple geometric designs.

☐ Activity 3: relating photo-cropping and sequences

☐ Measure directly or use proportions to find the widths.

☐ presentation

☐ Does your display include examples of both arithmetic and geometric sequences? What artists or work of art with which you are familiar best demonstrate the concepts of one-point perspective, two-point perspective, or vanishing points?

• Write the first four terms of an geometric sequence that has the length of the diagonal of the cropped area as its first term. Using the terms of your sequence as diagonal lengths, find the four corresponding photo widths. What do you notice about this list of widths? Check students’ work.

Finishing the Project

Scoring Rubric

The answers to the activities should help you complete your project. Prepare a presentation or demonstration that summarizes how an artist, a designer, or a photographer uses sequences. Present this information to your classmates. Then discuss the sequences you made.

4

Calculations, sequences, and formulas are correct. Drawings are neat, accurate, and clearly show the sequences. Explanations are thorough and well thought out.

3

Calculations, sequences, and formulas are mostly correct with some minor errors. Drawings are neat and mostly accurate. Explanations lack detail or are not completely accurate.

2

Calculations contain both minor and major errors. Drawings are not accurate.

1

Major concepts are misunderstood. Project satisfies few of the requirements and shows poor organization and effort.

0

Major elements of the project are incomplete or missing.

Reﬂect and Revise Review your summary. Are your drawings clear and correct? Are your sequences accurate? Practice your presentation in front of at least two people before presenting it to the class. Ask for their suggestions for improvement. Extending the Project Geometric and arithmetic patterns are used in other aspects of design and in other careers. Research other areas where sequences are applied.

Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.

Teacher’s Evaluation of the Project

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