# 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

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