Answers for 12 - Mrs. Andres-Mathlfandres.weebly.com/uploads/2/5/8/0/25801520/higher_alg_chapter_… · 12 16 20 24 28 32 36 67. a. 15 balls b. 35 balls c. Except for layer 1, there

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A4Algebra 2

Answer Transparencies for Checking Homework

59. true; i 5 1

! n

kai 5 (ka1 1 ka2 1

ka3 1 . . . 1 kan) 5 k(a1 1 a2 1

a3 1 . . . 1 an) 5 k i 5 1

! n

ai

60. true; i 5 1

! n

(ai 1 bi) 5

(a1 1 b1) 1 (a2 1 b2) 1

(a3 1 b3) 1 . . . 1 (an 1 bn) 5

(a1 1 a2 1 a3 1 . . . 1 an) 1

(b1 1 b2 1 b3 1 . . . 1 bn) 5

i 5 1

! n

ai 1 i 5 1

! n

bi

61. False. Sample answer:

i 5 1

! 4

(2i)(24i) ! 1 i 5 1

! 4

2i 2 1 i 5 1

! 4

24i 2 62. False. Sample answer:

i 5 1

! 4

(2i)2 ! 1 i 5 1

! 4

2i 2 2

12.1 Problem Solving

63. 608, 908, 1088, 1208, about 128.578; Tn 5 180(n 2 2); 18008

64. $50.50; 316 days. Sample answer: I used the special series formula for the sum of the fi rst n positive integers and set it equal to 50,000 (since there are 50,000 pennies in $500) and solved.

65. an 5 2n 2 1; 63 moves, 127 moves, 255 moves

66. a. about 1.6 astronomical units

b. about 239,360,000 km

c.

Position of planet from sun

Mea

n di

stan

ce fr

om s

un (a

. u.)

0 1 2 3 4 5 6 7 8 9 n

dn

048

12162024283236

67. a. 15 balls

b. 35 balls

c. Except for layer 1, there are always more balls in the same layer of the square pyramid. The difference in the number

of balls is n(n 2 1)

} 2 .

68.

Sn 5 1 } 2 1 n(n 1 1)(2n 1 1) }} 6 1

n(n 1 1) } 2 2

12.1 Mixed Review

69. 5 70. 4 71. 22

72. 5 } 4 73. 3 } 2 74. 2

Answers for 12.1 continuedFor use with pages 798–800

a2_mnlaect371566_c12at.indd A4a2_mnlaect371566_c12at.indd A4 9/2/09 11:23:07 PM9/2/09 11:23:07 PM

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A5Algebra 2


75. 23 76. 2

77. 23 78. 2 Ï}

10

79. 5 Ï}

2 80. Ï}

74

81. Ï}

34 82. Ï}

17

83. 5 84. Ï}

17

85. 2 Ï}

13 86. Ï}

205



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A1Algebra 2


12.1 Skill Practice

1. sigma notation

2. A sequence is a list of numbers, and a series is the sum of the terms of a sequence.

3. 3, 4, 5, 6, 7, 8

4. 5, 4, 3, 2, 1, 0

5. 1, 4, 9, 16, 25, 36

6. 3, 10, 29, 66, 127, 218

7. 1, 4, 16, 64, 256, 1024

8. 21, 24, 29, 216, 225, 236

9. 24, 21, 4, 11, 20, 31

10. 16, 25, 36, 49, 64, 81

11. 24, 22, 2 4 } 3 , 21, 2 4 } 5 , 2 2 } 3

12. 3, 3 } 2 , 1, 3 } 4 , 3 } 5 , 1 } 2

13. 2 } 3 , 1, 6 } 5 , 4 } 3 , 10 } 7 , 3 } 2

14. 1, 2 } 3 , 3 } 5 , 4 } 7 , 5 } 9 , 6 } 11

15. You can write the terms as 5(1) 2 4, 5(2) 2 4, 5(3) 2 4, 5(4) 2 4, a5 5 21, an 5 5n 2 4.

16. You can write the terms as 21 2 1, 22 2 1, 23 2 1, 24 2 1, a5 5 16, an 5 2n 2 1.

17. You can write the terms as (21)1(4 p 1), (21)2(4 p 2), (21)3(4 p 3), (21)4(4 p 4), a5 5 220, an 5 (21)n(4 p n).

18. You can write the terms as 13 1 1, 23 1 1, 33 1 1, 43 1 1, a5 5 126, an 5 n3 1 1.

19. You can write the terms as 2 } 3(1) ,

2 } 3(2) , 2 } 3(3) ,

2 } 3(4) , a5 5 2 } 15 , an 5 2 } 3n .

20. You can write the terms as 2(1)

} 1 1 2 ,

2(2)

} 2 1 2 , 2(3)

} 3 1 2 , 2(4)

} 4 1 2 , a5 5 10

} 7 ,

an 5 2n } n 1 2 .

21. You can write the terms as 1 } 4 , 2 } 4 , 3 } 4 ,

4 } 4 , 5 } 4 , a6 5 6 } 4 , an 5

n } 4 .

22. You can write the terms as

2(1) 2 1

} 1(10) , 2(2) 2 1

} 2(10) , 2(3) 2 1

} 3(10) ,

2(4) 2 1

} 4(10) , a5 5 9 } 50 , an 5

2n 2 1 } 10n .

23. You can write the terms as 0.7(1) 1 2.4, 0.7(2) 1 2.4, 0.7(3) 1 2.4, 0.7(4) 1 2.4, a5 5 5.9, an 5 0.7n 1 2.4.

Answers for 12.1For use with pages 798–800


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A2Algebra 2


24. You can write the terms as 5.8 2 1.6(1), 5.8 2 1.6(2), 5.8 2 1.6(3), 5.8 2 1.6(4), 5.8 2 1.6(5), a6 5 23.8, an 5 5.8 2 1.6n.

25. You can write the terms as 12 1 0.2, 22 1 0.2, 32 1 0.2, 42 1 0.2, a5 5 25.2, an 5 n2 1 0.2.

26. You can write the terms as 7.8(1) 1 1.2, 7.8(2) 1 1.2, 7.8(3) 1 1.2, 7.8(4) 1 1.2, a5 5 40.2, an 5 7.8n 1 1.2.

27. D

28. 2

n21

an

29.

n0 1 2 3 4 5 6 707

1421283542495663an

30.

3

n21

an

31.

4

n21

an

32.

4

n21

an



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A3Algebra 2


33.

4

n21

an

34.

5

n21

an

35.

1

n21

an

36.

1

n21

an

37. i 5 1

! 5

(3i 1 4) 38. i 5 1

! 5

(6i 1 4)

39. i 5 1

! `

(2i 2 3) 40. i 5 1

! `

(22)i

41. i 5 1

! `

(7i 2 4) 42. i 5 1

! 4

1 } 3i

43. i 5 1

! 7

i } 3 1 i 44. i 5 1

! `

(i2 2 2)

45. 42 46. 105 47. 100

48. 90 49. 82 50. 50

51. 761 } 140 52. 617 } 140 53. 35

54. 136 55. 325 56. 2109

57. The lower limit is zero, so the fi rst term should be 3; 3 1 5 1 7 1 9 1 11 1 13 5 48.

58. a. 1 } 8

b. 1 } n



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A6Algebra 2


12.2 Skill Practice

1. common difference

2. An arithmetic sequence is a list of numbers that have the same common difference between consecutive terms. An arithmetic series is the sum of the terms of the arithmetic sequence.

3. Arithmetic; there is a common difference of 3 between consecutive terms.

4. Not arithmetic; there is not a common difference between consecutive terms.

5. Arithmetic; there is a common difference of 9 between consecutive terms.


7. Arithmetic; there is a common difference of 0.5 between consecutive terms.




11. Arithmetic; there is a common difference of 1.5 between consecutive terms.

12. an 5 3n 2 2; 58

13. an 5 21 1 6n; 119

14. an 5 25 1 13n; 255

15. an 5 25 1 2n; 35

16. an 5 10 2 4n; 270

17. an 5 36 2 11n; 2184

18. an 5 2 2 } 3 1 2 } 3 n; 38 } 3

19. an 5 7 } 3 2 1 } 3 n; 2

13 } 3

20. an 5 20.6 1 2.1n; 41.4

21. The equation for an arithmetic sequence is not correct; an 5 a1 1 (n 2 1)d, an 5 37 1 (n 2 1)(213), an 5 50 2 13n.

22. The terms were substituted into the wrong places; 37 5 (n 2 1)(213), an 5 50 2 13n.

23. an 5 228 1 5n; points at (1, 223), (2, 218), (3, 213), (4, 28), (5, 23), (6, 2)



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A7Algebra 2


24. an 5 270 1 9n; points at (1, 261), (2, 252), (3, 243), (4, 234), (5, 225), (6, 216)

25. an 5 152 2 14n; points at (1, 138), (2, 124), (3, 110), (4, 96), (5, 82), (6, 68)

26. an 5 81 2 7n; points at (1, 74), (2, 67), (3, 60), (4, 53), (5, 46), (6, 39)

27. an 5 25 1 7 } 2 n; points at

(1, 21.5), (2, 2), (3, 5.5), (4, 9), (5, 12.5), (6, 16)

28. an 5 6 2 1 } 2 n; points at

(1, 5.5), (2, 5), (3, 4.5), (4, 4), (5, 3.5), (6, 3)

29. C

30. an 5 25 1 9n

31. an 5 9 1 5n

32. an 5 211 1 3n

33. an 5 22 2 4n

34. an 5 17 1 8n

35. an 5 13 1 2n

36. an 5 111 } 5 2 13

} 5 n

37. an 5 15

} 4 1 9 } 4 n

38. an 5 12 } 5 2 2 } 5 n

39. B

40. 175 41. 296 42. 2774

43. 2585 44. 252 45. 315

46. 450 47. 132 48. 161

49. an 5 23 1 5n

50. an 5 2 2 3n

51. an 5 21 2 2n

52. Sample answer: The graph of an is just points at every integer n and the graph of f (x) is a line. Both graphs have the same rate of change between points.

53. False. Sample answer: Doubling the common difference alone does not double the sum.

54. true; a 1 c 5 2b

55. 12 56. 8 57. 25

58. 5 59. 15 60. 9

61. 22,500 62. 2 } 3 , 2 8 } 3


63. a. an 5 6n

b. 271 cells



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A8Algebra 2


64. an 5 1 1 2n; 63 band members

65. a. an 5 24 1 8n

b. 576 blocks

66. a. n d(n)

1 16

2 48

3 80

4 112

b. an 5 216 1 32n

66. c.

12

n21

d(n)

67. $100

68. a. n dn (in.) ln (in.)

1 2 2!2 2.008 2.008!3 2.016 2.016!4 2.024 2.024!

b. arithmetic; ln 5 [2 1 0.008(n 2 1)]!

c. 375 times; about 4119 in., or 343 ft

d. Sample answer: A roll with a diameter of 7 in. requires 625 wraps and contains 8828 in. of paper. This is about 2.14 times as much paper as the 5 in. roll, so you would expect a cost of about 2.14($1.50) = $3.21.

69. a1 5 2y

} n 2 x

12.2 Mixed Review

70. 16,807 71. 2216, 216

72. 232, 32 73. 3 3 Ï}

9

74. 615 75. 259

76. 3 77. about 2.153

78. about 0.314 79. about 0.029

80. 13 81. 1 } 2

82. 6 2 } 3 , 6, 6

83. about 40.4, 43, 43

84. 84.625, 82.5, 92

85. about 21.29, 22, none

86. about 2.6, 2.6, 1.9

87. about 3.6, 3.8, none

88. 23 hats



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A9Algebra 2


12.3 Skill Practice

1. common ratio

2. When you divide consecutive terms you have the same ratio.

3. Not geometric; there is no common ratio.

4. Geometric; there is a common ratio of 4.

5. Geometric; there is a common

ratio of 1 } 6 .




9. Geometric; there is a common

ratio of 1 } 2 .






15. an 5 (24)n 2 1; 4096

16. an 5 6(3)n 2 1; 4374

17. an 5 4(6)n 2 1; 186,624

18. an 5 7(25)n 2 1; 109,375

19. an 5 2 1 3 } 4 2 n 2 1; 729 } 2048

20. an 5 3 1 2 2 } 5 2 n 2 1; 192 } 15,625

21. an 5 4 1 1 } 2 2 n 2 1; 1 } 16

22. an 5 20.3(22)n 2 1; 219.2

23. an 5 22(0.4)n 2 1; 20.008192

24. an 5 7(20.6)n 2 1; 0.326592

25. an 5 5(22.8)n 2 1; 2409.45152

26. an 5 120(1.5)n 2 1; 1366.875

27. B

28. an 5 5(3)n 2 1

125

n21

an



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A10Algebra 2


29. an 5 22(6)n 2 1

1600

n1

an

30. an 5 3(2)n 2 1

10

n21

an

31. an 5 30 1 1 } 2 2 n 2 1

3

n1

an

32. an 5 4096 1 1 } 8 2 n 2 1

400

n21

an



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A11Algebra 2


33. an 5 768 1 2 1 } 4 2 n 2 1

100

n21

an

34. an 5 3(5)n 2 1

1000

n21

an

35. an 5 2(4)n 2 1

200

n21

an

36. an 5 4(5)n 2 1

1500

n21

an

37. The exponent should be n 2 1 instead of n; an 5 3(2)n 2 1.

38. r and a1 are switched around in the formula; an 5 a1rn 2 1,

an 5 3(2)n 2 1.

39. an 5 3(2)n 2 1 or an 5 3(22)n 2 1

40. an 5 1(5)n 2 1 or an 5 1(25)n 2 1

41. an 5 1 2 1 } 4 2 (4)n 2 1



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A12Algebra 2


42. an 5 1 10 } 9 2 (3)n 2 1

43. an 5 280 1 1 } 2 2 n 2 1 or

an 5 80 1 2 1 } 2 2 n 2 1

44. an 5 6(24)n 2 1

45. an 5 6(3)n 2 1

46. an 5 7 1 1 } 2 2 n 2 1 or an 5 7 1 2 1 } 2 2 n 2 1

47. an 5 32

} 27 1 3 3 Ï}

12 } 4 2 n 2 1

48. 5115 49. 131,070

50. 255 } 32 51. 1365 } 256

52. 527,345

} 256 53. 838,861

54. C

55. The graph of f(x) is a curve defi ned for all real numbers, and the graph of an is each point from the graph of f(x) for positive integer values of x.

56. a. S5 5 1 1 2 x5 } 1 2 x 2

b. S4 5 3x 1 1 2 16x8 }

1 2 2x2 2 12.3 Problem Solving

57. a. an 5 5(2)n 2 1

b. 75 skydivers

58. a. an 5 32 1 1 } 2 2 n 2 1; 1 ! n ! 6

b. 63 games

59. a. an 5 512 1 1 } 2 2 n 2 1

b. 10. Sample answer: After the 10th pass, there is only 1 term to choose from so it must be the answer.

60. a. an 5 (8)n 2 1; 2,396,745 squares

b. an 5 8 } 9 1 8 } 9 2 n 2 1

; about 0.2433

61. a. an 5 19,000 1 1000n, arithmetic; bn 5 20,000(1.04)n 2 1, geometric

b.

3000

n1

an

c. Company A: $590,000; Company B: about $595,562

d. 19 yr

62. $132,877.70



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A13Algebra 2


12.3 Mixed Review

63.

10 221

13

29

12

56

64.

2 3 4 51023 22 21

472 6

65.

2 31023 22 212627 25 24

1142 8 2.721.8

66. 2 5 } 9 67. 7 } 10

68. 8 } 13 69. 0, 9

70. 29, 22 71. 16

72. 210 73. 378

74. 9 75. 333

76. 2128 77. 1084

12.1–12.3 Mixed Review of Problem Solving

1. a. an 5 45,000(1.035)n 2 1

b. $51,638.54

c. $2,323,020.48

2. a. an 5 (2n 2 1)!

b. i 5 1

! n

(2i 2 1)!

c. !, 4!, 16 !; it quadruples the area.

3. 2 1 4n; arranging the tables with their short ends together creates room for 4 more chairs with each table that is added, where arranging the tables with their long ends together creates room for 2 more chairs with each table that is added.

4. Sample answer: i 5 1

! 8

19 } 14 1 23

} 14 i

5. 105 pieces of chalk;

501

6. an 5 2 1 7n; 72 in.; change the formula to be an 5 2 1 7(n 2 1)



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A14Algebra 2


7. a. Geometric; there is a constant

ratio of 1 } 2 between terms.

b. an 5 66 1 1 } 2 2 n 2 1

c.

7

n

an

1

exponential decay

d. 14 h

8. Sample answer: 3, 6, 9, 12, 15;

45 } 31 , 90 } 31 , 180 } 31 , 360 } 31 , 720 } 31



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A15Algebra 2


12.4 Skill Practice

1. partial sum

2. If !r! < 1, then the series has a sum.

3. S1 5 0.5, S2 ø 0.67, S3 ø 0.72, S4 ø 0.74, S5 ø 0.75; Sn appears to be approaching 0.75.

0.1

x

y

1

4. S1 5 0.67, S2 5 1, S3 ø 1.17, S4 5 1.25, S5 ø 1.29; Sn appears to be approaching 1.3.

0.15

x

y

1

5. S1 5 4, S2 5 6.4, S3 5 7.84, S4 ø 8.71, S5 ø 9.22; Sn appears to be approaching 10.

1

x

y

1

6. S1 5 0.25, S2 5 1.5, S3 5 7.75, S4 5 39, S5 5 195.25; Sn continues to increase.

20

x

y

1

7. 10 8. no sum

9. no sum 10. 88 } 15

11. 12 } 5 12. 2 25

} 3

13. 63 } 17 14. no sum



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A16Algebra 2


15. no sum 16. 2 8 } 5

17. 7 } 10 18. no sum

19. Since r > 1, the infi nite geometric series has no sum.

20. 2 3 } 8 21. 1 } 2

22. no sum 23. 18

24. 2 } 9 25. 4 } 9

26. 16 } 99 27. 625 } 999

28. 3200 } 99 29. 130,000

} 999

30. 1 } 11 31. 5 } 18

32. C

33. 0.9 } 1 2 0.1 5 0.9

} 0.9 5 1


! `

2.5 1 1 } 2 2 i 2 1,

i 5 1

! `

10 } 3 1 1 } 3 2 i 2 1

35. 2 1 } 4 < x < 1 } 4 ; S 5 1 } 1 2 4x

36. 24 < x < 4; S 5 6 }

1 2 1 } 4 x


37. 70 ft

38. $2,916,666.67; s 5 350,000

} 1 2 0.88

39. D

40. Sd 5 40 ft, St 5 2 sec; Yes; the total distance traveled is 40 feet and it occurs after 2 seconds.

41. a. 12 ft; 9 ft

b. i 5 1

! `

12(0.75)i 2 1

c. 56 ft

d. 2(0.75h)

} 1 2 0.75 1 h 5 7h

42. a. an 5 3n 2 1

} 4n

b. 1; eventually no area remains

12.4 Mixed Review

43. 61% 44. 0.46 45. 0.1

46. an 5 23 1 5n

47. an 5 235 1 8n

48. an 5 216 2 18n

49. an 5 41 2 7n

50. an 5 59.5 1 6.5n

51. an 5 17.5 2 1.5n

52. an 5 4(2.5)n 2 1

53. an 5 6(23)n 2 1

54. an 5 10,368 1 2 1 } 4 2 n 2 1



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A17Algebra 2


55. an 5 1 } 9 (6)n 2 1

56. an 5 10,240

} 81 (0.75)n 2 1

57. an 5 2(4)n 2 1



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A18Algebra 2


12.5 Skill Practice

1. iteration

2. An explicit rule gives the value based on the position of the term in the sequence while a recursive rule gives the value based on previous term(s) in the sequence.

3. 1, 4, 7, 10, 13

4. 4, 8, 16, 32, 64

5. 21, 26, 211, 216, 221

6. 3, 2, 22, 211, 227

7. 2, 5, 26, 677, 458,330

8. 4, 6, 26, 666, 443,546

9. 2, 8, 10, 18, 22

10. 2, 4, 2, 22, 24

11. 2, 3, 6, 18, 108

12. A

13. a1 5 21, an 5 an 2 1 2 7

14. a1 5 3, an 5 4an 2 1

15. a1 5 4, an 5 23an 2 1

16. a1 5 1, an 5 an 2 1 1 7

17. a1 5 44, an 5 1 } 4 an 2 1

18. a1 5 1, a2 5 4, an 5 an 2 2 1 an 2 1

19. a1 5 54, an 5 an 2 1 2 11

20. a1 5 3, a2 5 5, an 5 an 2 2 p an 2 1

21. a1 5 16, a2 5 9, an 5 an 2 2 2 an 2 1

22. When writing a recursive rule, you must defi ne the previous information needed; a1 5 5, a2 5 2, an 5 an 2 2 2 an 2 1.

23. The rule does not work for all of the terms of the sequence; a1 5 5, a2 5 2, an 5 an 2 2 2 an 2 1.

24. 4, 10, 28 25. 24, 214, 264

26. 3, 25, 27 27. 22, 24, 25

28. 9, 11, 12 1 } 3 29. 5, 21, 437

30. 3, 19, 723 31. 2, 4, 14

32. 28, 2208, 2130,208

33. C

34. a1 5 3, a2 5 8, an 5 (an 2 2)2 1 an 2 1

35. a1 5 1, a2 5 2, an 5 4(an 2 2 1 an 2 1)

36. a1 5 5, an 5 Ï}

3 an 2 1

37. a1 5 2, a2 5 5, an 5 3an 2 2 1 an 2 1

38. a1 5 8, a2 5 4, an 5 an 2 2 } an 2 1



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A19Algebra 2


39. a1 5 23, a2 5 22, an 5 21(an 2 2 1 an 2 1)

40. Sample answer: a1 5 2, a2 5 4, a3 5 7, an 5 an 2 3 1 an 2 2 1 an 2 1,

2, 4, 7, 13, 24, 44, 81, 149

41. Sample answer: If the fi rst two iterates are 2, the third iterate must also be 2.

42. a. 5, 18, 9, 30, 15, 48, 24, 12, 6, 3

b. Sample answer: a1 5 2: 2, 1, 6, 3, 12, 6, 3, 12, 6, 3; a1 5 3: 3, 12, 6, 3, 12, 6, 3, 12, 6, 3; a1 5 6: 6, 3, 12, 6, 3, 12, 6, 3, 12, 6; the terms of the sequence will eventually repeat the numbers 3, 12, 6.


43. a. a1 5 5000, an 5 0.8an 2 1 1 500;

3524 fi sh

b. The population of the lake approaches 2500 fi sh.

44. a1 5 34, an 5 0.6an 2 1 1 16; the amount of chlorine in the pool approaches 40 ounces.

45. a1 5 2000, an 5 1.014an 2 1 2 100; 24 mo.

Sample answer: As long as Gladys does not add anything to her credit card and continues her payments, her 24th payment will only be $62.14.

46. 1, 1, 2, 3, 5

47. a. a1 5 20, an 5 0.7an 2 1 1 20

b. 66 2 } 3 mg

c. The maintenance level of the drug doubles as well; a1 5 40, an 5 0.7(an 2 1) 1 40

48 a. an 5 1.08an 2 1 2 30,000

b. an 2 1 5 an 1 30,000

} 1.08 ;

a0 5 about 294,544.42

12.5 Mixed Review

49. 3 Ï}

2 cm 50. 2 Ï}

14 ft

51. 9 m 52. 64

53. 9 54. 1 } 8

12.4–12.5 Mixed Review of Problem Solving

1. a. i 5 1

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16.8(0.7)i 2 1

b. 68 ft


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A20Algebra 2


2. a. 1, 2, 4, 8, 16, 32

b. geometric

c. an 5 2n 2 1 and a1 5 1, an 5 2an 2 1

3. 60; 06

4. Sample answer: an 5 22 1 5n and a1 5 3, an 5 an 2 1 1 5

5. Sample answer: The sum continues to grow larger because the terms of the sequence are constantly growing larger and never approach any specifi c value.

6. Finite; the common ratio is less than 1; 160 in.

7. a. 0.54%; a1 5 10,000, an 5 1.0054an 2 1 2 196

b. $8244.47

c. 47 months

d. Sample answer: Yes; by paying an extra $50 each month, you are paying the loan off early and therefore will pay less interest.

8. 5000 trees; 0005


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2 1 1 } 2 2 i 2 1



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Answers for 12 - Mrs. Andres-Mathlfandres.weebly.com/uploads/2/5/8/0/25801520/higher_alg_chapter_… · 12 16 20 24 28 32 36 67. a. 15 balls b. 35 balls c. Except for layer 1, there