aat solutions - ch12

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    667Algebra 2

    Worked-Out Solution Key

    Prerequisite Skills (p. 792)

    1. The domain of f (x) is all real numbers except x 0.

    2. The range of g(x) is all real numbers.

    3. The composition f (g(x)) is equal to 1 } 4x 1 2 .

    4. 7x 1 3 5 31 5. 9 5 2x 2 7

    7x 5 28 16 5 2x

    x 5 4 8 5 x

    6. 14 5 23x 1 8 7. 10 2 3x 5 28

    6 5 23x 23x 5 18

    22 5 x x 5 26

    8. 11x 1 9 5 3x 1 17 9. 2x 1 3 5 26 2 x

    8x 5 8 3x 5 29

    x 5 1 x 5 23

    10. 3x 1 y 5 0 3 4 12x 1 4y 5 0

    22x 2 4y 5 230 22x 2 4y 5 230

    10x 5 230

    x 5 23

    3(23) 1 y 5 0 y 5 9 The solution is (23, 9).

    11. 2x 2 2y 5 10 2x 2 2y 5 10

    x 1 y 5 210 3 2 2x 1 2y 5 220

    4x 5 210

    x 5 2 5 } 2

    2 5 } 2 1 y 5 210 y 5 2

    15 } 2

    The solution is 1 2 5 } 2 , 2 15 } 2 2 . 12. 4x 2 5y 5 25 4x 2 5y 5 25

    0.5x 1 1.5y 5 18.5 3 28 24x 2 12y 5 2148

    217y 5 2123

    y 5 123

    } 17

    4x 2 5 1 123 } 17 2 5 25 x 5 260 } 17 The solution is 1 260 } 17 , 123 } 17 2 . 13. f (g(x)) 5 2(22x21) 2 1 5 24x21 2 1 Domain: all real numbers except x 5 0

    14. f ( f (x)) 5 2(2x 2 1) 2 1 5 4x 2 3

    Domain: all real numbers

    15. g(g(x)) 5 22(22x21)21 5 x Domain: all real numbers except x 5 0

    Lesson 12.1

    12.1 Guided Practice (pp. 794797)

    1. a1 5 1 1 4 5 5 2. f (1) 5 (22)1 2 1 5 1

    a2 5 2 1 4 5 6 f (2) 5 (22)2 2 1 5 22

    a3 5 3 1 4 5 7 f (3) 5 (22)3 2 1 5 4

    a4 5 4 1 4 5 8 f (4) 5 (22)4 2 1 5 28

    a5 5 5 1 4 5 9 f (5) 5 (22)5 2 1 5 16

    a6 5 6 1 4 5 10 f (6) 5 (22)6 2 1 5 232

    3. a1 5 1 } 1 1 1 5

    1 } 2

    a2 5 2 } 2 1 1 5

    2 } 3

    a3 5 3 } 3 1 1 5

    3 } 4

    a4 5 4 } 4 1 1 5

    4 } 5

    a5 5 5 } 5 1 1 5

    5 } 6

    a6 5 6 } 6 1 1 5

    6 } 7

    4. The sequence 3, 8, 15, 24, . . .

    5

    n21

    an

    can be written as 1 p 3, 2 p 4, 3 p 5, 4 p 6, . . .. The next term is a5 5 5 p 7 5 35. A rule for the nth term is an 5 n(n 1 2).

    5. a9 5 92 5 81

    There are 81 apples in the 9th layer.

    6. ai 5 5i

    Lower limit 5 1

    Upper limit 5 20

    Summation notation: i 5 1

    20

    5i

    7. ai 5 i2 }

    i2 1 1

    Lower limit 5 1

    Upper limit 5 in nity

    Summation notation: i 5 1

    `

    i2 }

    i2 1 1

    8. ai 5 6i

    Lower limit 5 1

    Upper limit 5 in nity

    Summation notation: i 5 0

    `

    6i

    9. 4 1 i

    Lower limit 5 1

    Upper limit 5 8

    Summation notation: i 5 1

    8

    (4 1 i)

    Chapter 12

    n2ws-1200-a.indd 667 6/27/06 11:30:51 AM

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    668Algebra 2Worked-Out Solution Key

    10. i 5 1

    5

    8i 5 8(1) 1 8(2) 1 8(3) 1 8(4) 1 8(5)

    5 8 1 16 1 24 1 32 1 40

    5 120

    11. k 5 3

    7

    (k2 2 1) 5 (32 2 1) 1 (42 2 1) 1 (52 2 1) 1

    (62 2 1) 1 (72 2 1) 5 8 1 15 1 24 1 35 1 48

    5 130

    12. i 5 1

    34

    1 5 34

    13. n 5 1

    6

    n 5 1 1 2 1 3 1 4 1 5 1 6 5 21

    14. n 5 1

    9

    n2 5 12 1 22 1 32 1 42 1 52 1 62 1 72 1 82 1 92

    5 1 1 4 1 9 1 16 1 25 1 36 1 49 1 64 1 81

    5 285

    There are 285 apples in the stack.

    12.1 Exercises (pp. 798800)

    Skill Practice

    1. Another name for summation notation is sigma notation.

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

    3. a1 5 1 1 2 5 3 4. a1 5 6 2 1 5 5

    a2 5 2 1 2 5 4 a2 5 6 2 2 5 4

    a3 5 3 1 2 5 5 a3 5 6 2 3 5 3

    a4 5 4 1 2 5 6 a4 5 6 2 4 5 2

    a5 5 5 1 2 5 7 a5 5 6 2 5 5 1

    a6 5 6 1 2 5 8 a6 5 6 2 6 5 0

    5. a1 5 12 5 1 6. f (1) 5 13 1 2 5 3

    a2 5 22 5 4 f (2) 5 23 1 2 5 10

    a3 5 32 5 9 f (3) 5 33 1 2 5 29

    a4 5 42 5 16 f (4) 5 43 1 2 5 66

    a5 5 52 5 25 f (5) 5 53 1 2 5 127

    a6 5 62 5 36 f (6) 5 63 1 2 5 218

    7. a1 5 41 2 1 5 1 8. a1 5 21

    2 5 21

    a2 5 42 2 1 5 4 a2 5 22

    2 5 24

    a3 5 43 2 1 5 16 a3 5 23

    2 5 29

    a4 5 44 2 1 5 64 a4 5 24

    2 5 216

    a5 5 45 2 1 5 216 a5 5 25

    2 5 225

    a6 5 46 2 1 5 1024 a6 5 26

    2 5 236

    9. f (1) 5 12 2 5 5 24 10. a1 5 (1 1 3)2 5 16

    f (2) 5 22 2 5 5 21 a2 5 (2 1 3)2 5 25

    f (3) 5 32 2 5 5 4 a3 5 (3 1 3)2 5 36

    f (4) 5 42 2 5 5 11 a4 5 (4 1 3)2 5 49

    f (5) 5 52 2 5 5 20 a5 5 (5 1 3)2 5 64

    f (6) 5 62 2 5 5 31 a6 5 (6 1 3)2 5 81

    11. f (1) 5 2 4 } 1 5 24 12. a1 5 3 } 1 5 3

    f (2) 5 2 4 } 2 5 22 a2 5 3 } 2

    f (3) 5 2 4 } 3 a3 5 3 } 3 5 1

    f (4) 5 2 4 } 4 5 21 a4 5 3 } 4

    f (5) 5 2 4 } 5 a5 5 3 } 5

    f (6) 5 2 4 } 6 5 2 2 } 3 a6 5

    3 } 6 5

    1 } 2

    13. a1 5 2(1)

    } 1 1 2 5 2 } 3 14. f (1) 5

    1 } 2(1) 2 1 5 1

    a2 5 2(2)

    } 2 1 2 5 1 f (2) 5 2 } 2(2) 2 1 5

    2 } 3

    a3 5 2(3)

    } 3 1 2 5 6 } 5 f (3) 5

    3 } 2(3) 2 1 5

    3 } 5

    a4 5 2(4)

    } 4 1 2 5 4 } 3 f (4) 5

    4 } 2(4) 2 1 5

    4 } 7

    a5 5 2(5)

    } 5 1 2 5 10

    } 7 f (5) 5 5 } 2(5) 2 1 5

    5 } 9

    a6 5 2(6)

    } 6 1 2 5 3 } 2 f (6) 5

    6 } 2(6) 2 1 5

    6 } 11

    15. Given terms: 5 p 1 2 4, 5 p 2 2 4, 5 p 3 2 4, 5 p 4 2 4, . . .

    Next term: 5 p 5 2 4 5 21 Rule for nth term: 5n 2 4

    16. Given terms: 21 2 1, 22 2 1, 23 2 1, 24 2 1, . . .

    Next term: 25 2 1 5 16

    Rule for nth term: 2n 2 1

    17. Given terms: (21)1(4 p 1), (21)2(4 p 2), (21)3(4 p 3), (1)4(4 p 4), . . .

    Next term: (21)5(4 p 5) 5 220 Rule for nth term: (21)n(4n)

    18. Given terms: 13 1 1, 23 1 1, 33 1 1, 43 1 1, . . .

    Next term: 53 1 1 5 126

    Rule for nth term: n3 1 1

    19. Given terms: 2 } 3 p 1 , 2 } 3 p 2 ,

    2 } 3 p 3 , 2 } 3 p 4 , . . .

    Next term: 2 } 3 p 5 5 2 } 15

    Rule for nth term: 2 } 3n

    20. Given terms: 1 p 2 } 1 1 2 , 2 p 2 } 2 1 2 ,

    3 p 2 } 3 1 2 ,

    4 p 2 } 4 1 2 , . . .

    Next term: 5 p 2

    } 5 1 2 5 10

    } 7

    Rule for nth term: 2n } 2 1 n

    21. Given terms: 1 } 4 , 2 } 4 ,

    3 } 4 ,

    4 } 4 , 5 } 4 , . . .

    Next term: 6 } 4

    Rule for nth term: n } 4

    Chapter 12, continued

    n2ws-1200-a.indd 668 6/28/06 1:48:37 PM

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    669Algebra 2

    Worked-Out Solution Key

    Chapter 12, continued22. Given terms: 2 p 1 2 1 } 1 p 10 ,

    2 p 2 2 1 } 2 p 10 , 2 p 3 2 1

    } 3 p 10 , 2 p 4 2 1 } 4 p 10 , . . .

    Next term: 2 p 5 2 1

    } 5 p 10 5 9 } 50

    Rule for nth term: 2n 2 1

    } 10n

    23. Given terms: 2.4 1 0.7(1), 2.4 1 0.7(2), 2.4 1 0.7(3), 2.4 1 0.7(4), . . .

    Next term: 2.4 1 0.7(5) 5 5.9

    Rule for nth term: 2.4 1 0.7n

    24. Given terms: 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), . . .

    Next term: 5.8 2 1.6(6) 5 23.8

    Rule for nth term: 5.8 2 1.6n

    25. Given terms: 0.2 1 12, 0.2 1 22, 0.2 1 32, 0.2 1 42, . . .

    Next term: 0.2 1 52 5 25.2

    Rule for nth term: 0.2 1 n2

    26. Given terms: 1.2 1 7.8(1), 1.2 1 7.8(2), 1.2 1 7.8(3), 1.2 1 7.8(4), . . .

    Next term: 1.2 1 7.8(5) 5 40.2

    Rule for nth term: 1.2 1 7.8n

    27. D;

    a1 5 1(1 1 1)

    } 2 5 1

    a2 5 2(2 1 1)

    } 2 5 3

    a3 5 3(3 1 1)

    } 2 5 6

    a4 5 4(4 1 1)

    } 2 5 10

    28.2

    n21

    an 29.

    n1

    7

    an

    30.

    3

    n21

    an

    31. an

    4

    n21

    32.

    4

    n21

    an 33.

    4

    n21

    an

    34.

    5

    n21

    an 35.

    1

    n21

    an

    36.

    1

    n21

    an

    37. ai 5 3i 1 4

    Lower limit 5 1

    Upper limit 5 5

    Summation notation: i 5 1

    5

    (3i 1 4)

    38. 6i 2 1

    Lower limit 5 1

    Upper limit 5 5

    Summation notation: i 5 1

    5

    (6i 2 1)

    39. 2i 2 3

    Lower limit 5 1

    Upper limit 5 in nity

    Summation notation: i 5 1

    `

    (2i 2 3)

    n2ws-1200-a.indd 669 6/28/06 1:49:13 PM

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    670Algebra 2Worked-Out Solution Key

    40. (22)i

    Lower limit 5 1

    Upper limit 5 in nity

    Summation notation: i 5 1

    `

    (22)i

    41. 7i 2 4

    Lower limit 5 1

    Upper limit 5 in nity

    Summation notation: i 5 1

    `

    (7i 2 4)

    42. 1 } 3i

    Lower limit 5 1