Chapter 11 Sequences Induction And Probability 111 Sequences

Ch.11 Sequences,Induction,andProbability

11.1 SequencesandSummationNotation

1 FindParticularTermsofaSequencefromtheGeneralTerm

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writethefirstfourtermsofthesequencewhosegeneraltermisgiven.

1) an=3n

A) 3

,

6

,

9

,

12 B) 0,3

,

6

,

9C)4

,

5

,

6

,

7D)2

,

1

,

0

,

–1

2) an=n–4

A) –3

,

–2

,

–1

,

0B)

–4

,

–3

,

–2

,

–1C)

–1

,

0

,

1

,

2D)

–16

,

–12

,

–8

,

–4

3) an=2n–1

A) 1

,

3

,

5

,

7B)1

,

2

,

3

,

4C)3

,

5

,

7

,

9D)

–1

,

–3

,

–5

,

–7

4) an=4(3n–2)

A) 4

,

16

,

28

,

40 B) 1

,

4

,

7

,

10 C) –8

,

4

,

16

,

28 D) 4

,

8

,

12

,

16

5) an=3n

A) 3

,

9

,

27

,

81 B) 1

,

8

,

27

,

64 C) 1

,

3

,

9

,

27 D) 9

,

27

,

81

,

243

6) an=2

3

n

A) 2

3,4

9,8

27 ,16

81 B) 1,2

3,4

9,8

27 C) 2

3,2

6,2

9,2

12 D) 1,4

9,8

27 ,16

81

7) an=(–4)n

A) –4

,

16

,

–64

,

256 B) –4

,

–16

,

–64

,

–256 C) 4

,

–16

,

–64

,

–256 D) 4

,

–16

,

64

,

–256

8) an=–1

3

n

A) –1

3,1

9,–1

27 ,1

81 B) –1

3,–1

9,–1

27 ,–1

81

C) –1

3,1

6,–1

9,–1

12 D) 1

3,–1

6,1

9,–1

12

9) an=(–1)n(n+8)

A) –9

,

10

,

–11

,

12 B) –9

,

–10

,

–11

,

–12 C) –9

,

–20

,

–33

,

–48 D) 9

,

10

,

11

,

12

10) an=(–1)n+1(n+8)

A) 9

,

–10

,

11

,

–12 B) –9

,

10

,

–11

,

12 C) 9

,

–20

,

33

,

–48 D) –10

,

11

,

–12

,

13

11) an=n+2

2n–1

A) 3,4

3,1,6

7B) 3,–4

3,1,6

7C) –3,4

3,1,6

7D) –3,–4

3,1,6

7

Page1

12) an=(–1)n+1

n+3

A) 1

4,–1

5,1

6,–1

7B) –1

4,1

5,–1

6,1

7C) 1

4,–1

10 ,1

18 ,–1

28 D) –1

5,1

6,–1

7,1

8

13) an=3

n2

A) 3,3

4,3

9,3

16 B) 1,2

4,3

9,4

16 C) 3

4,3

9,3

16 ,3

25 D) 1,1

4,1

9,1

16

Solvetheproblem.

14) Adepositof$6000ismadeinanaccountthatearns9%interestcompoundedquarterly.Thebalanceinthe

accountafternquartersisgivenbythesequence

an=6000 1+0.09

4

nn=1,2,3,…

Findthebalanceintheaccountafter8years.

A) $12,228.62 B) $12,301.62 C) $12,295.62 D) $12,088.62

15) Adepositof$7000ismadeinanaccountthatearns9%interestcompoundedquarterly.Thebalanceinthe

accountafternquartersisgivenbythesequence

an=7000 1+0.09

4

nn=1,2,3,…

Findthebalanceintheaccountafter36quarters.

A) $15,594.71 B) $15,692.71 C) $15,677.71 D) $15,413.71

16) Adepositof$6000ismadeinanaccountthatearns5.6%interestcompoundedquarterly.Thebalanceinthe

accountafternquartersisgivenbythesequence

an=6000 1+0.056

4

n,n=1,2,3,…

Findthebalanceintheaccountafter4years.

A) $7494.77 B) $4996.52 C) $6343.12 D) $3762.52

2 UseRecursionFormulas

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writethefirstfourtermsofthesequencedefinedbytherecursionformula.

1) a1=1andan=an–1–3forn≥2

A) 1

,

–2

,

–5

,

–8B)

–3

,

–6

,

–9

,

–12 C) 1

,

4 ,7 ,10 D) 1

,

0 ,–3 ,–6

2) a1=–3andan=an–1–2forn≥2

A) –3

,

–5

,

–7

,

–9B)3

,

1

,

–1

,

–3C)3

,

5

,

7

,

9D)

–3

,

–3

,

–1

,

1

3) a1=2andan=4an–1forn≥2

A) 2

,

8

,

32

,

128 B) 2

,

7

,

6

,

5C)4

,

16

,

64

,

128 D) 2

,

10

,

34

,

130

4) a1=–3andan=–2an–1forn≥2

A) –3

,

6

,

–12

,

24 B) 3

,

–6

,

12

,

–24 C) –3

,

–6

,

–12

,

–24 D) –3

,

8

,

–14

,

26

5) a1=5andan=3an–1–4forn≥2

A) 5

,

11

,

29

,

83 B) 5

,

11

,

41

,

131 C) 5

,

15

,

45

,

135 D) 5

,

19

,

61

,

187

Page2

3 UseFactorialNotation

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writethefirstfourtermsofthesequencewhosegeneraltermisgiven.

1) an=n4

(n+1)!

A) 1

2,8

3,27

8,32

15 B) 2,4

3,1

2,2

15 C) 1

2,8

3,27

4,64

5D) 2,4

3,1,4

5

2) an=(n+1)!

n3

A) 2,3

4,8

9,15

8B) 2

3,1,8

3,10 C) 2,3

4,4

9,5

16 D) 2

3,1,4

3,5

3

3) an=–2(n+2)!

A) –12

,

–48

,

–240

,

–1440 B) –12

,

96

,

–720

,

5760

C) –4

,

–24

,

–144

,

–960 D) –4

,

12

,

–48

,

240

4) an= 3n

(n+1)!

A) 3

2,3

2,9

8,27

40 B) 3

2,3,27

4,81

5C) 2

3,3,27

4,5

81 D) 3

2,3

2,9

4,27

20

5) an=–4(n+1)!

n!

A) –8

,

–12

,

–16

,

–20 B) –8,–6,–8

3,–5

6C) –3

,

–2

,

–1

,

0D)

–8,–6,–16

3,–5

Evaluatethefactorialexpression.

6) 7!

5!

A) 42 B) 2! C) 7

5D) 7

7) 6!

8!

A) 1

56 B) 56 C) 2! D) 1

2!

8) 8!

6!2!

A) 28 B) 8 C) 0! D) 1

9) 8!

7!

A) 8 B) 1 C) 8

7D) 8!

Page3

10) 12!

6!6!

A) 924 B) 1848 C) 665,280 D) 462

11) (n+8)!

n+8

A) (n+7)! B) 1 C) 8! D) n+8!

12) n(n+3)!

(n+4)!

A) n

n+4B) n

4C) 1

n+4D) n

(n+4)!

4 UseSummationNotation

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheindicatedsum.

1)

7

i=4

7i

∑

A) 154 B) 49 C) 77 D) 105

2)

5

i=1

(i–6)

∑

A) –15 B) –1C)

–6D)

–14

3)

6

i=3

(3i–3)

∑

A) 42 B) 36 C) 45 D) 24

4)

4

i=1

1

9i

∑

A) 25

108 B) 1

36 C) 5

36 D) 11

54

5)

10

i=7

1

i–2

∑

A) 533

840 B) 487

1260 C) –800

2401 D) 26

6)

4

i=1

2i

∑

A) 30 B) 18 C) 14 D) 20

Page4

7)

5

i=3

(i2–10)

∑

A) 20 B) 5 C) –6D)

–18

8)

4

k=2

k(k+3)

∑

A) 56 B) 60 C) 38 D) 27

9)

4

k=1

(–1)k(k+3)

∑

A) 2 B) –22 C) 22 D) 14

10)

8

i=5

4

∑

A) 16 B) 104 C) 12 D) 84

11)

4

i=1

(–1

4)i

∑

A) –51

256 B) 51

256 C) –47

256 D) 85

256

12)

5

i=1

(–1)i–1

(i–1)!

∑

A) 3

8B) –3

8C) 5

12 D) –5

12

13)

6

i=3

i!

(i–1)!

∑

A) 18 B) 10 C) 3 D) 6

14)

5

i=1

(i+2)!

(i+1)!

∑

A) 25 B) 129

20 C) 19 D) 377

60

Expressthesumusingsummationnotation.Use1asthelowerlimitofsummationandifortheindexofsummation.

15) 3+12+27+...+75

A)

5

i=1

3i2

∑B)

5

i=0

3i2

∑C)

5

i=1

i2

∑D)

5

i=1

32i

∑

Page5

16) 1

4+2

5+1

2+...+4

5

A)

12

i=1

i

i+3

∑B)

12

i=0

i

i+3

∑C)

n

i=1

i

i+3

∑D)

12

i=3

i

i+1

∑

17) 2+4+6+...+10

A)

5

i=1

2i

∑B)

5

i=0

2i

∑C)

5

i=1

i2

∑D)

5

i=1

2i2

∑

18) 62+123+184+...+489

A)

8

i=1

(6i)i+1

∑B)

8

i=1

(6i)i

∑C)

8

i=1

2(i–1)i+1

∑D)

8

i=1

6i2i–1

∑

19) a+1+a+2

2+...+a+4

4

A)

4

i=1

a+i

i

∑B)

4

i=0

a+i

i

∑C)

n

i=0

a+i

i

∑D)

n

i=1

a+i

i

∑

20) a+ar+ar2+...+ar13

A)

14

i=1

ari–1

∑B)

13

i=1

ari

∑C)

13

i=1

(ar)i

∑D)

13

i=1

(ar)i–1

∑

Expressthesumusingsummationnotation.Usealowerlimitofsummationnotnecessarily1andkfortheindexof

summation.

21) 5+6+7+8+...+22

A)

19

k=2

(k+3)

∑B)

22

k=5

(k+3)

∑C)

25

k=8

(k+3)

∑D)

17

k=1

(k+3)

∑

22) 15+19+23+27+...+47

A)

10

k=2

4k+7

∑B)

32

k=0

4k+7

∑C)

10

k=1

4k+7

∑D)

32

k=2

4k+7

∑

23) 5

6+6

7+7

8+8

9+...+15

16

A)

15

k=5

k

k+1

∑B)

15

k=6

k+1

k

∑C)

15

k=5

k+1

k

∑D)

15

k=6

k

k+1

∑

24) 4+9

2+5+11

2+...+10

A)

20

k=8

k

2

∑B)

20

k=1

k

2

∑C)

12

k=8

k

2

∑D)

20

k=2

k

2

∑

Page6

25) a+ar+ar2+...+ar11

A)

11

k=0

ark

∑B)

12

k=1

ark

∑C)

11

k=0

(ar)k

∑D)

11

k=1

ark

∑

26) (a+1)+(a+b)+(a+b2)+...+(a+bn)

A)

n

k=0

(a+bk)

∑B)

n

k=1

(a+bk)

∑C)

n–1

k=0

(a+bk)

∑D)

n

k=0

abk

∑

Solvetheproblem.

27) Thebargraphbelowshowsacompanyʹsyearlyprofitsfrom1991to1999.Letanrepresentthecompanyʹs

profit,inmillions,inyearn,wheren=1correspondsto1991,n=2correspondsto1992,andsoon.

Find

7

i=3

ai

∑

A) $356.9million B) $400.7 million C) $142.6 million D) $371.5 million

28) Thefinitesequencewhosegeneraltermis

an=0.19n2–1.04n+7.48

wheren=1,2,3,…,9modelsthetotaloperatingcosts,inmillionsofdollars,foracompanyfrom1991through

1999.

Find

4

i=1

ai

∑

A) $25.22million B) $26.52 million C) $32.25 million D) $33.15 million

11.2 ArithmeticSequences

1 FindtheCommonDifferenceforanArithmeticSequence

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findthecommondifferenceforthearithmeticsequence.

1) 6

,

10

,

14

,

18

,

...

A) 4 B) 12 C) 3 D) 6

2) 4

,

5

,

6

,

7

,

...

A) 1 B) 3 C) –1D)

–3

3) 2

,

0

,

–2

,

–4

,

...

A) –2B)

–6C)

–4D)6

Page7

4) 566

,

558

,

550

,

542

,

...

A) –8 B) 8 C) 566 D) –566

2 WriteTermsofanArithmeticSequence

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writethefirstfivetermsofthearithmeticsequence.

1) a1=9;d=3

A) 9

,

12

,

15

,

18

,

21 B) 12

,

15

,

18

,

21

,

24 C) 0,9

,

12

,

15

,

18 D) 9

,

11

,

13

,

15

,

17

2) a1=7;d=–1

A) 7

,

6

,

5

,

4

,

3B)0,7

,

6

,

5

,

4C)

–7

,

–6

,

–5

,

–4

,

–3D)11

,

9

,

7

,

5

,

3

3) a1=13;d=–4

A) 13

,

9

,

5

,

1

,

–3B)17

,

13

,

9

,

5

,

1C)9

,

5

,

1

,

–3

,

–7D)13

,

9

,

4

,

1

,

–3

4) a1=–21;d=2

A) –21

,

–19

,

–17

,

–15

,

–13 B) –17

,

–15

,

–13

,

–11

,

–9

C) –13

,

–15

,

–17

,

–19

,

–21 D) –17

,

–19

,

–21

,

–23

,

–25

5) a1=–1

,

d=–1

A) –1

,

–2

,

–3

,

–4

,

–5B)

–1

,

0

,

1

,

2

,

3

C) –1

,

–1

,

–1

,

–1

,

–1D)

–1,0,1

3,1

2,3

5

6) a1=–1

2;d=–3

2

A) –1

2,–2,–7

2,–5,–13

2B) –1

2,1, 5

2,4, 11

2

C) –1

2,–1,–3

2,–2,–5

2D) –1

2,–2, 5

2,–5,11

2

7) an=an–1+5;a1=9

A) 9

,

14

,

19

,

24

,

29 B) 8

,

13

,

18

,

23

,

28 C) 5

,

14

,

23

,

32

,

41 D) 9

,

5

,

14

,

19

,

24

8) an=an–1–3;a1=–20

A) –20

,

–23

,

–26

,

–29

,

–32 B) –21

,

–24

,

–27

,

–30

,

–33

C) –3

,

–23

,

–43

,

–63

,

–83 D) –20

,

–3

,

–23

,

–26

,

–29

9) an=an–1+3.8;a1=12

A) 12

,

15.8

,

19.6

,

23.4

,

27.2 B) 11

,

14.8

,

18.6

,

22.4

,

26.2

C) 3.8

,

15.8

,

27.8

,

39.8

,

51.8 D) 12

,

3.8

,

15.8

,

19.6

,

23.4

10) an=an–1–3.8;a1=–18

A) –18

,

–21.8

,

–25.6

,

–29.4

,

–33.2 B) –19

,

–22.8

,

–26.6

,

–30.4

,

–34.2

C) –3.8

,

–21.8

,

–39.8

,

–57.8

,

–75.8 D) –18

,

–3.8

,

–21.8

,

–25.6

,

–29.4

Page8

11) an=an–1–1

5;a1=–2

5

A) –2

5,–3

5,–4

5,–1,–6

5B) –2

5,–1

5,0,1

5,2

5

C) –2

5,–4

5,–6

5,–8

5,–2D)

–2

5,–3

5,–4

5,–1

3 UsetheFormulafortheGeneralTermofanArithmeticSequence

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usetheformulaforthegeneralterm(thenthterm)ofanarithmeticsequencetofindtheindicatedtermofthesequence

withthegivenfirstterm,a1,andcommondifference,d.

1) Finda8whena1=8

,

d=–3.

A) –13 B) –16 C) 29 D) 32

2) Finda26 whena1=6

,

d=3.

A) 81 B) 84 C) –69 D) –72

3) Finda26 whena1=–5,d=–1

5.

A) –10 B) –51

5C) 0 D) 1

5

4) Finda17whena1=21

,

d=–6.

A) –75 B) –81 C) –96 D) 117

5) Finda60whena1=9

,

d=–5.

A) –286 B) –291 C) 304 D) 309

Writeaformulaforthegeneralterm(thenthterm)ofthearithmeticsequence.Thenusetheformulaforantofinda20

,

the20thtermofthesequence.

6) 2

,

6

,

10

,

14

,

18

,

...

A) an=4n–2;a20=78 B) an=4n–1;a20 =79

C) an=2n–1;a20=39 D) an=2n–4;a20 =36

7) 1

,

5

,

9

,

13

,

17

,

...

A) an=4n–3;a20=77 B) an=3n–4;a20 =56

C) an=4n+3;a20=83 D) an=n+4;a20 =24

8) 19

,

10,1

,

–8

,

...

A) an=28–9n;a20=–152 B) an=19 –9n;a20=–161

C) an=9n–28;a20=152 D) an=9n–19;a20=161

9) –13

,

–20,–27

,

–34

,

...

A) an=–7n–6;a20=–146 B) an= –7n–13;a20=–153

C) an=7n+6;a20=146 D) an=7n+13;a20=153

Page9

10) a1=7

5,d=2

5

A) an=2

5n+1;a20=9B)a

n=2

5n+7

5;a20=47

5

C) an=7

5n–1;a20=27 D) an=7

5n+2

5;a20=142

5

11) a1=–7

,

d=0.2

A) an=0.2n–7.2;a20=–3.2 B) an=0.2n–7;a20=–3

C) an=–7n+7.2;a20=–132.8 D) an= –7n+0.2;a20=–139.8

12) an=an–1+6

,

a1=18

A) an=6n+12

,

a20=132 B) an=18n–12

,

a20=348

C) an=6n+24

,

a20=144 D) an=12n+6

,

a20 =246

13) an=an–1–7

,

a1=32

A) an=–7n+39

,

a20=–101 B) an=32n+39

,

a20=679

C) an=–7n+25

,

a20=–115 D) an=25n+32

,

a20=532

Solvetheproblem.

14)

J

acieisconsideringajobthatoffersamonthlystartingsalaryof$2500 andguaranteesheramonthlyraiseof

$160duringherfirstyearonthejob.Findthegeneraltermofthisarithmeticsequenceandhermonthlysalary

attheendofherfirstyear.

A) an=2340+160n;$4260 B) an=2500 +160n;$4420

C) an=2340+160(n–1);$4100 D) an=2500 +160n;$4260

15) Totrainforarace,Willbeginsbyjogging15 minutesonedayperweek.Heincreaseshisjoggingtimeby5

minuteseachweek.Writethegeneraltermofthisarithmeticsequence,andfindhowmanywholeweeksit

takesforhimtoreachajoggingtimeofonehour.

A) an=5n+10;10weeks B) an=5n+15;9 weeks

C) an=5n+10;9weeks D) an=5n+15;10 weeks

16) Thepopulationofatownisincreasingby500 inhabitantseachyear.Ifitspopulationatthebeginningof1990

was29,421,whatwasitspopulationatthebeginningof1998?

A) 33,421inhabitants B) 235,228 inhabitants

C) 32,921inhabitants D) 470,456 inhabitants

4 UsetheFormulafortheSumoftheFirstnTermsofanArithmeticSequence

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheindicatedsum.

1) Findthesumofthefirst60termsofthearithmeticsequence:16

,

22

,

28

,

34

,

...

A) 11,580 B) 11,588 C) 376 D) 11,760

2) Findthesumofthefirst60termsofthearithmeticsequence:13

,

5

,

–3

,

–11

,

...

A) –13,380 B) –13,375 C) –467 D) –13,620

3) Findthesumofthefirst60termsofthearithmeticsequence:–11

,

–7

,

–3

,

1

,

...

A) 6420 B) 6428 C) 229 D) 6540

Page10

4) Findthesumofthefirst50termsofthearithmeticsequence:–14

,

–21

,

–28

,

–35

,

...

A) –9275 B) –9267 C) –364 D) –9450

5) Findthesumofthefirst55termsofthearithmeticsequence:2

,

4

,

6

,

8

,

...

A) 3080 B) 3086 C) 112 D) 3135

6) Find2+4+6+8+...,thesumofthefirst100 positiveeven integers.

A) 10,100 B) 10,104 C) 9999 D) 9995

7) Findthesumoftheoddintegersbetween126 and62.

A) 3008 B) 3013 C) 6204 D) 3018

8) Findthesumoftheevenintegersbetween25 and55.

A) 600 B) 560 C) 680 D) 640

Writeoutthefirstthreetermsandthelasttermofthearithmeticsequence.

9)

13

i=1

(4i+1)

∑

A) 5+9+13+...+53 B) 1 +5+9+...+53

C) 1–5+9–...+53 D) 5 +17 +49 +...+209

10)

55

i=1

–3i

∑

A) –3–6–9–...–165 B) –3+9–27 +...–495

C) –3–9–27–...–495 D) –1–3–6–...–165

Usetheformulaforthesumofthefirstntermsofanarithmeticsequencetofindtheindicatedsum.

11)

49

i=1

(4i+6)

∑

A) 5194 B) 5096 C) 5390 D) 5561.5

12)

36

i=1

(5i–6)

∑

A) 3114 B) 3024 C) 3222 D) 3348

13)

28

i=1

(–6i+7)

∑

A) –2240 B) –2156 C) –2100 D) –2002

14)

28

i=1

(–5i–6)

∑

A) –2198 B) –2128 C) –2058 D) –1960

15)

60

i=1

–4i

∑

A) –7320 B) –7442 C) –7198 D) –7316

Page11

Solvetheproblem.

16) Atheaterhas30rowswith25seatsinthefirstrow,29 inthesecondrow,33 inthethirdrow,andsoforth.How

manyseatsareinthetheater?

A) 2490seats B) 2550 seats C) 4980 seats D) 5100 seats

17) Abrickstaircasehasatotalof14stepsThebottomsteprequires123 bricks.Eachsuccessivesteprequires5

fewerbricksthanthepriorone.Howmanybricksarerequiredtobuildthestaircase?

A) 1267bricks B) 2177 bricks C) 2534 bricks D) 1232 bricks

18) Apersonputs$23intoabankaccountonJanuary1,$28 onFebruary1,$33 onMarch1,andsoforth.How

muchhasthepersonputintothebankaccountbyDecember30?

A) $606 B) $636 C) $1212 D) $468

19) Anewexhibitisscheduledtoopenatthelocalmuseum.Museumofficialsexpectthat10,000peoplewillvisit

theexhibitinitsfirstweek,andthatthenumberofvisitorswilldropby30peopleperweekafterthefirstweek

duringthefirst6months.Findthetotalnumberofvisitorsexpectedintheexhibitʹsfirst7weeks.

A) 69,370visitors B) 69,190 visitors C) 59,550 visitors D) 49,550 visitors

20) Aspartofherretirementsavingsplan,Patriciadeposited$150 inabankaccountduringherfirstyearinthe

workforce.Duringeachsubsequentyear,shedeposited$45morethanthepreviousyear.Findhowmuchshe

depositedduringhertwentiethyearintheworkforce.Findthetotalamountdepositedinthetwentyyears.

A) $1005;$11,550 B) $1050;$12,000 C) $1005;$23,100 D) $1050;$24,000

11.3 GeometricSequencesandSeries

1 FindtheCommonRatioofaGeometricSequence

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Ifthegivensequenceisageometricsequence,findthecommonratio.

1) 4

,

12

,

36

,

108

,

324

A) 3 B) 1

3

C) 12 D) notageometricsequence

2) 3

,

–9

,

27

,

–81

,

243

A) –3B)3

C) –12 D) notageometricsequence

3) 4

3,8

3,16

3,32

3,64

3

A) 2 B) notageometricsequence

C) 6 D) 4

4) 3

4,3

16 ,3

64 ,3

256 ,3

1024

A) 1

4B) 4 C) 1

40 D) 40

Page12

5) 1

8,1

12 ,1

16 ,1

20

A) notageometricsequence B) 4

C) 4 1

4D) 1

4

2 WriteTermsofaGeometricSequence

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writethefirstfivetermsofthegeometricsequence.

1) a1=2;r=3

A) 2

,

6

,

18

,

54

,

162 B) 6

,

18

,

54

,

162

,

486 C) 2

,

5

,

8

,

11

,

14 D) 3

,

6

,

12

,

24

,

48

2) a1=5;r=1

3

A) 5,5

3,5

9,5

27 ,5

81 B) 5

,

15

,

45

,

135

,

405 C) 5

3,5

9,5

27 ,5

81 ,5

243 D) 5,16

3,17

3,6,19

3

3) a1=4;r=–3

A) 4

,

–12

,

36

,

–108

,

324 B) 4

,

12

,

36

,

–108

,

324

C) –3

,

–12

,

36

,

–108

,

324 D) 4

,

1

,

–2

,

–5

,

–8

4) a1=–7;r=–4

A) –7

,

28

,

–112

,

448

,

–1792 B) –7

,

–28

,

–112

,

448

,

–1792

C) –4

,

28

,

–112

,

448

,

–1792 D) –7

,

–11

,

–15

,

–19

,

–23

5) an=8an–1;a1=3

A) 3

,

24

,

192

,

1536

,

12,288 B) 24

,

192

,

1536

,

12,288

,

98,304

C) 8

,

24

,

192

,

1536

,

12,288 D) 3

,

11

,

19

,

27

,

35

6) an=6an–1;a1=–2

A) –2

,

–12

,

–72

,

–432

,

–2592 B) –12

,

–72

,

–432

,

–2592

,

–15,552

C) 6

,

12

,

–72

,

–432

,

–2592 D) –2

,

4

,

10

,

16

,

22

7) an=–8an–1;a1=3

A) 3

,

–24

,

192

,

–1536

,

12,288 B) –24

,

192

,

–1536

,

12,288

,

–98,304

C) –8

,

–24

,

–192

,

–1536

,

–12,288 D) 3

,

–5

,

–13

,

–21

,

–29

8) an=–8an–1;a1=–5

A) –5

,

40

,

–320

,

2560

,

–20,480 B) 40

,

–320

,

2560

,

–20,480

,

163,840

C) –8

,

–40

,

–320

,

–2560

,

–20,480 D) –5

,

–13

,

–21

,

–29

,

–37

3 UsetheFormulafortheGeneralTermofaGeometricSequence

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usetheformulaforthegeneralterm(thenthterm)ofageometricsequencetofindtheindicatedtermofthesequence

withthegivenfirstterm,a1,andcommonratio,r.

1) Finda6whena1=8

,

r=5.

A) 25,000 B) 125,000 C) 3125 D) 200

Page13

2) Finda5whena1=8

,

r=–3.

A) 648 B) –1944 C) 81 D) –648

3) Finda10whena1=4

,

r=2.

A) 2048 B) 4096 C) 2052 D) 22

4) Finda10whena1=–4

,

r=2.

A) –2048 B) –4096 C) –2044 D) 14

5) Finda11whena1=3

,

r=–2.

A) 3072 B) –6144 C) 3076 D) –17

6) Finda12whena1=–5

,

r=–3.

A) 885,735 B) –2,657,205 C) 885,739 D) –38

7) Finda6whena1=9600,r=–1

2.

A) –300 B) –150 C) 300 D) –30

8) Finda11whena1=3000,r=1

2.

A) 375

128 B) 375

256 C) 375

512 D) 3005

9) Finda8whena1=2

,

000,000,r=0.1.

A) 0.2 B) 0.02 C) 2 D) 0.002

10) Finda8whena1=40,000,r=–0.1.

A) –0.004 B) 0.0004 C) 0.004 D) –0.04

Writeaformulaforthegeneralterm(thenthterm)ofthegeometricsequence.

11) 4

,

8

,

16

,

32

,

64

,

...

A) an=4(2)n–1B) an=4(2)nC) an=a1+2nD) an=4(2n)

12) –8

,

–16

,

–32

,

–64

,

–128

,

...

A) an=–8(2)n–1B) an=–8(2)nC) an=a1+2nD) an= –8(2n)

13) 2

,

–6

,

18

,

–54

,

162

,

...

A) an=2(–3)n–1B) an=2(–3)nC) an=a1–3nD) an=2(–3n)

14) 2,1

2,1

8,1

32 ,1

128 ,...

A) an=21

4

n–

1

B) an=21

4

nC) an=21

4

n+

1

D) an=21

16

n–

1

15) 4,–4

3,4

9,–4

27 ,4

81 ,...

A) an=4–1

3

n–

1

B) an=4–1

3

nC) an=4–1

3

n+

1

D) an=4–1

9

n–

1

Page14

16) 1

2,–1

18 ,1

162 ,–1

1458 ,...

A) an=1

2–1

9

n–

1

B) an=1

2–1

9(n–1) C) an=1

9–1

2

n–

1

D) an=1

2

n–

1

–5

9

17) –1

6,1

42 ,–1

294 ,1

2058 ,...

A) an=–1

6–1

7

n–

1

B) an=–1

6–1

7(n–1)

C) an=–1

7–1

6

n–

1

D) an=–1

6

n–

1

+4

21

18) 0.00009

,

0.0018

,

0.036

,

0.72

,

...

A) an=0.00009(20)n–1B) an=9(0.2)n

C) an=9(20)nD) an=0.0009(20)n–1

Thegeneraltermofasequenceisgiven.Determinewhetherthegivensequenceisarithmetic,geometric,orneither.If

thesequenceisarithmetic,findthecommondifference;ifitisgeometric,findthecommonratio.

19) an=3n–2

A) arithmetic,d=3 B) geometric,r=3 C) arithmetic,d= –2 D) neither

20) an=5n

A) geometric,r=5 B) arithmetic,d=5 C) geometric,r=6 D) neither

21) an=5

4

n

A) geometric,r=5

4B) arithmetic,d=5

4C) geometric,r=4

5D) neither

22) an=4n2–3

A) arithmetic,d=–3 B) geometric,r=4 C) arithmetic,d=4 D) neither

23) an=3n2

A) geometric,r=3 B) geometric,r=3

2C) arithmetic,d=3 D) neither

Solvetheproblem.

24) Afootballplayersignsacontractwithastartingsalaryof$810,000 peryearandanannualincreaseof4.5%

beginninginthesecondyear.Whatwilltheathleteʹssalarybe,tothenearestdollar,intheseventhyear?

A) $1,054,831 B) $1,055,748 C) $1,053,605 D) $1,056,838

25) Keyanatakesajobwithastartingsalaryof$30,000 forthefirstyearwithanannualincreaseof4%beginningin

thesecondyear.WhatisKeyanaʹssalary,tothenearestdollar,intheseventhyear?

A) $37,960 B) $38,703 C) $36,519 D) $40,035

26) Ms.PattersonproposestogiveherdaughterClaireanallowanceof$0.20 onthefirstdayofher12–day

vacation,$0.40onthesecondday,$0.80onthethirdday,andsoon.FindtheallowanceClairewouldreceive

onthelastdayofhervacation.

A) $409.60 B) $819.20 C) $2048.20 D) $2.40

Page15

27) Apendulumbobswingsthroughanarc80 incheslongonitsfirstswing.Eachswingthereafter,itswingsonly

80%asfarasonthepreviousswing.Whatisthelengthofthearcafter6swings?Roundtotwodecimalplaces.

A) 26.21inches B) 20.97 inches C) 16.78 inches D) 320.00 inches

28) Thefollowingtableshowsacountryʹspopulationfrom1995to1998:

Year 1995 1996 1997 1998

Populationinmillions 11.80 12.51 13.26 14.06

Dividethepopulationforeachyearbythepopulationintheprecedingyear.Usethisratiotowritethegeneral

termofthegeometricsequencedescribingthecountryʹspopulationgrowthnyearsafter1994.Thenestimate

thecountryʹspopulation,inmillions,in2005.

A) an=11.80(1.06)n–1;21.13million B) an=11.80(1.06)n–1;22.4million

C) an=11.80(1.05)n–1;25.48million D) an=11.80(1.05)n–1;23.59million

4 UsetheFormulafortheSumoftheFirstnTermsofaGeometricSequence

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usetheformulaforthesumofthefirstntermsofageometricsequencetosolve.

1) Findthesumofthefirstfourtermsofthegeometricsequence:2

,

10

,

50

,

....

A) 312 B) 62 C) 19 D) 156

2) Findthesumofthefirst8termsofthegeometricsequence:5

,

10

,

20

,

40

,

80

,

....

A) 1275 B) 1255 C) 1312 D) 1277

3) Findthesumofthefirstfourtermsofthegeometricsequence:–2

,

–10

,

–50

,

....

A) –312 B) –62 C) 312 D) –156

4) Findthesumofthefirst12termsofthegeometricsequence:–3

,

–9

,

–27

,

–81

,

–243

,

....

A) –797,160 B) –797,180 C) –797,123 D) –797,158

5) Findthesumofthefirst14termsofthegeometricsequence:5

,

–15

,

45

,

–135

,

405

,

....

A) –5,978,710 B) –5,978,712 C) –5,978,703 D) –5,978,716

6) Findthesumofthefirstfivetermsofthegeometricsequence: 2

3,8

3,32

3,....

A) 682

3B) 226 C) 682

15 D) 226

5

7) Findthesumofthefirstfivetermsofthegeometricsequence: 3

1,3

2,3

4,....

A) 93

16 B) 3

8C) 93 D) 3

4

8) Findthesumofthefirst9termsofthegeometricsequence: 1

5,3

5,9

5,27

5,81

5,....

A) 9841

5B) 9839

5C) 9848

5D) 1967

Page16

9) Findthesumofthefirst13termsofthegeometricsequence: 1

3,–2

3,4

3,–8

3,16

3,....

A) 2731

3B) 2729

3C) 2738

3D) 2725

3

Findtheindicatedsum.Usetheformulaforthesumofthefirstntermsofageometricsequence.

10)

7

i=1

3

4

i

∑

A) 42,591

16,384 B) 14,197

4096 C) 3367

4096 D) 10,101

16,384

11)

5

i=1

3·3i

∑

A) 1089 B) 1845 C) 90 D) 117

12)

5

i=1

4(–3)i

∑

A) –732 B) –2460 C) 69 D) 240

13)

5

i=1

4

3·4i

∑

A) 5456

3B) 5435

3C) 5396

3D) 5399

3

14)

3

i=1

2

5

i+1

∑

A) 156

625 B) 78

125 C) 32

125 D) 812

625

Solvetheproblem.

15) Asmallbusinessownermade$60,000thefirstyearheownedhisstoreandmadeanadditional3% overthe

previousyearineachsubsequentyear.Findhowmuchhemadeduringhisfourthyearofbusiness.Findhis

totalearningsduringthefirstfouryears.(Roundtothenearestcent,ifnecessary.)

A) $65,563.62;$251,017.62 B) $1.62;$61,855.62

C) $131,820.00;$371,220.00 D) $171,366.00;$542,586.00

16) AsSuneeimprovesheralgebraskills,shetakes0.9 timesaslongtocompleteeachhomeworkassignmentasshe

tooktocompletethepreceedingassignment.Ifittookher55minutestocompleteherfirstassignment,find

howlongittookhertocompletethefifthassignment.Findthetotaltimeshetooktocompleteherfirstfive

homeworkassignments.(Roundtothenearestminute.)

A) 36min;225min B) 32min;225 min C) 36 min;189 min D) 32min;189 min

17) Ajobpaysasalaryof27,000thefirstyear.Duringthenext5 years,thesalaryincreasesby5%eachyear.What

isthesalaryforthe6thyear?Whatisthetotalsalaryoverthe6–yearperiod?(Roundtothenearestcent.)

A) $34,459.60;$183,651.65 B) $34,459.60;$149,192.04

C) $36,182.58;$219,834.23 D) $36,182.58;$149,212.04

Page17

5 FindtheValueofanAnnuity

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.Roundtothenearestdollarifneeded.

1) Tosaveforretirement,youdecidetodeposit$2000 intoanIRAattheendofeachyearforthenext30 years.If

theinterestrateis6%peryearcompoundedannually,findthevalueoftheIRAafter30years.

A) $158,116 B) $9487 C) $147,280 D) $1,978,136

2) Lookingaheadtoretirement,yousignupforautomaticsavingsinafixed–income401Kplanthatpays6%per

yearcompoundedannually.Youplantoinvest$2500attheendofeachyearforthenext20years.Howmuch

willyouraccounthaveinitattheendof20years?

A) $91,964 B) $93,734 C) $90,421 D) $93,262

3) Lonniedeposits$100eachmonthintoanaccountpayingannualinterestof6%compoundedmonthly.How

muchwillhisaccounthaveinitattheendof15years?

A) $29,082 B) $2328 C) $28,928 D) $29,211

4) Laurainvests$200eachquarterinafixed–interestmutualfundpayingannualinterestof7%compounded

quarterly.Howmuchwillheraccounthaveinitattheendof10years?

A) $11,447 B) $2763 C) $34,463 D) $11,576

6 UsetheFormulafortheSumofanInfiniteGeometricSeries

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findthesumoftheinfinitegeometricseries,ifitexists.

1) 3+3

4+3

16 +3

64 +...

A) 4 B) 15

4C) 3

4D) doesnotexist

2) 5–5

4+5

16 –5

64 +...

A) 4 B) 15

4C) –5

4D) doesnotexist

3) 18+6+2+2

3+...

A) 27 B) –9 C) 26 D) doesnotexist

4) –12–4–4

3–4

9–...

A) –18 B) 6 C) –52

3D) doesnotexist

5) 1

2–2+8–...

A) 1

10 B) 32,768 C) –8192 D) doesnotexist

Page18

6)

∞

i=1

4(–0.7)i–1

∑

A) 40

17 B) 40

3C) –40

17 D) –40

3

Expresstherepeatingdecimalasafractioninlowestterms.

7) 0.6= 6

10 +6

100 +6

1,000 +6

10,000 …

A) 2

3B) 2

33 C) 33

50 D) 333

500

8) 0.88= 88

100 +88

10,000 +88

1,000,000 +…

A) 8

9B) 8888

999 C) 8800

999 D) 1111

1250

9) 0.2

A) 2

9B) 20

9C) 1

5D) 1

50

10) 0.46

A) 46

99 B) 4600

99 C) 23

50 D) 23

500

11) 0.825

A) 275

333 B) 95

111 C) 33

40 D) 57

200

Solvetheproblem.

12) Apendulumbobswingsthroughanarc40 incheslongonitsfirstswing.Eachswingthereafter,itswingsonly

70%asfarasonthepreviousswing.Howfarwillitswingaltogetherbeforecomingtoacompletestop?Round

tothenearestinchwhennecessary.

A) 133inches B) 57inches C) 67 inches D) 114 inches

11.4 MathematicalInduction

1 UnderstandthePrincipleofMathematicalInduction

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

AstatementSnaboutthepositiveintegersisgiven.WritestatementsS1

,

S2

,

andS3

,

andshowthateachofthese

statementsistrue.

1) Sn:1+4+7+...+(3n–2)=n(3n–1)

2

2) Sn:12+42+72+...+(3n–2)2=n(6n2–3n–1)

2

3) Sn:1·2+2·3+3·4+...+n(n+1)=n(n+1)(n+2)

3

Page19

4) Sn:2isafactorofn2+3n

AstatementSnaboutthepositiveintegersisgiven.WritestatementsSkandSk+1

,

simplifyingSk+1completely.

5) Sn:4+9+14+...+(5n–1)=n(5n+3)

2

6) Sn:12+42+72+...+(3n–2)2=n(6n2–3n–1)

2

7) Sn:1·2+2·3+3·4+...+n(n+1)=n(n+1)(n+2)

3

8) Sn:2isafactorofn2+11n

2 ProveStatementsUsingMathematicalInduction

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Usemathematicalinductiontoprovethatthestatementistrueforeverypositiveintegern.

1) 6+12+18+...+6n=3n(n+1)

2) 1·7+2·7+3·7+...+7n=7n(n+1)

2

3) 2+5+8+...+(3n–1)=n(3n+1)

2

4) 3+3

4+3

16 +...+3

4n–1=41–1

4n

5) 2isafactorofn2–n+2

11.5 TheBinomialTheorem

1 EvaluateaBinomialCoefficient

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Evaluatethegivenbinomialcoefficient.

1) 9

6

A) 84 B) 504 C) 42 D) 1

2) 6

2

A) 15 B) 6 C) 0 D) 1

3) 10

5

A) 252 B) 504 C) 30,240 D) 126

Page20

4) 4

1

A) 4 B) 1 C) 4

3D) 4!

5) 9

9

A) 1 B) 362,880 C) 2 D) 0

6) 294

2

A) 43,071 B) 86,142 C) 294!

292! D) 292

7) 130

128

A) 8385 B) 65

64 C) 1,073,280 D) 8385

128

2 ExpandaBinomialRaisedtoaPower

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UsetheBinomialTheoremtoexpandthebinomialandexpresstheresultinsimplifiedform.

1) (x+4)3

A) x3+12x2+48x+64 B) x3+4x2+16x+64

C) 3x+12 D) x3+64

2) (4x+5)3

A) 64x3+240x2+300x+125 B) 64x3+240x2+240x+125

C) 16x6+20x3+15,625 D) 16x2+40x+25

3) (x+2y)3

A) x3+6x2y+12xy2+8y3B) x3+2x2y+4xy+4xy2+8y2+8y3

C) 3x+6y D) x3+8y3

4) (4x–2y)3

A) 64x3–96x2y+48xy2–8y3B) 64x3–32x2y+16xy2–8y3

C) 16x3y–8x2y2+4xy3D) 16x3y–16x2y2+4xy3

5) (x–4)4

A) x4–16x3+96x2–256x+256 B) x4–4x3+96x2–128x+256

C) x4–16x3–128x2–256x+256 D) x4–16x3+96x2+16x+256

6) (2x+3)4

A) 16x4+96x3+216x2+216x+81 B) 48x4+288x3+216x2+648x+81

C) 16x4+81x4D) 16x3+96x2+216x+216

Page21

7) (x2+5y)4

A) x8+20x6y+150x4y2+500x2y3+625y4B) x8+5x6y+150x4y2+250x2y3+625y4

C) x4+20x3y+150x2y2+500xy3+625y4D) x8+20x6y+150x4y2+20x2y3+625y4

8) (x5+2y)4

A) x20+8x15y+24x10y2+32x5y3+16y4B) x9+6x8y+24x7y2+24x5y3+16y4

C) x9+8x8y+12x7y2+8x5y3+2y4D) x20+6x15y+24x10y2+24x5y3+16y4

9) (x+8)5

A) x5+40x4+640x3+5120x2+20,480x+32,768 B) x5+40x4+640x3+5120x2+20,480x+8

C) x5+40x4+1280x3+10,240x2+20,480x+32,768 D) x5+40x4+1280x3+10,240x2+20,480x+8

10) (2x+4)5

A) 32x5+320x4+1280x3+2560x2+2560x+1024 B) 32x5+64x4+128x3+256x2+512x+1024

C) (4x2+16x+16)5D) 32x5+2560x4+2560x3+2560x2+2560x+1024

11) (x+5y)5

A) x5+25x4y+250x3y2+1250x2y3+3125xy4+3125y5

B) x5+25x4y+250x3y2+1250x2y3+3125xy4+3125

C) x5+25x4y+500x3y2+2500x2y3+3125xy4+3125y5

D) x5+25x4y+500x3y2+2500x2y3+3125xy4+5y5

12) (x+2y)6

A) x6+12x5y+60x4y2+160x3y3+240x2y4+192xy5+64y6

B) x6+12x5y+48x4y2+144x3y3+192x2y4+192xy5+64y6

C) x6+12x5y+64x4y2+40x3y3+64x2y4+12xy5+2y6

D) x6+12x5y+24x4y2+36x3y3+24x2y4+12xy5+2y6

Solvetheproblem.

13) Acompanymodelsitsyearlyexpensesinmillionsofdollarsusingtheequationf(t)=0.03t3–0.7t2+1.14t+3.9

wheret=0represents1989.Thecompanyʹsaccountmanagerdecidestoadjustthemodelsothatt=0

correspondsto1999ratherthan1989.Todothis,sheobtainsg(t)=f(t+10).UsetheBinomialTheoremto

expressg(t)indescendingpowersoft.

A) 0.03t3+0.2t2–3.86t–24.7 B) 0.03t3–1.6t2+24.14t+1.04

C) 0.03t3+0.2t2+24.14t+8.6 D) 0.03t3–1.6t2–6.86t+22.6

3 FindaParticularTerminaBinomialExpansion

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Writethefirstthreetermsinthebinomialexpansion,expressingtheresultinsimplifiedform.

1) (x+2) 18

A) x 18 +36 x 17 +612 x 16 B) x 18 +36 x 17 +1224 x 16

C) x 18 +34 x 17 +612 x 16 D) x 18 +34 x 17 +1224 x 16

2) (x+6)

20

A) x20+120 x19+6840 x18 B) x20+1140 x19+6840 x18

C) x20+6840 x19+6840 x18 D) x20+120 x19–6840 x18

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3) (x–2)

17

A) x17–34 x16+544 x15 B) x17–34 x16–544 x15

C) x17+34 x16+544 x15 D) x17+34 x16–544 x15

4) ( 3 x+3y)2

A) 9 x 2+18 x 1y+9x0y2B) 2 x 2+1x1y+0x0y2

C) 9 x 2y+18 x 1y2+9x0y3D) 2 x 2y+1x1y2+0x0y3

5) (x2+6)

9

A) x18+54 x16+1296 x14 B) x18+54 x16+2592 x14

C) x18+60 x16+1296 x14 D) x18+60 x16+2592 x14

Findthetermindicatedintheexpansion.

6) (x+3y)8;7thterm

A) 20,412x2y6B) 6804x2y7C) 6804x6y2D) 20,412x6y2

7) (x–3y)11;8thterm

A) –721,710x4y7B) 240,570x4y8C) 240,570x7y4D) –721,710x7y4

8) (3x+2y)10;7thterm

A) 1,088,640x4y6B) 544,320x4y7C) 544,320x6y4D) 362,880x6y4

9) (x2+y3)8;3rdterm

A) 28x12y6B) 28x8y5C) 3360x12y6D) 3360x8y5

10) (5x+4)5;5thterm

A) 6400x B) 8000x2C) 5120 D) 1600x

11) (3x+4)5;5thterm

A) 3840x B) 2880x2C) 5120 D) 960x

11.6 CountingPrinciples,Permutations,andCombinations

1 UsetheFundamentalCountingPrinciple

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) Inhowmanywayscan7playersbeassignedto7 positionsonabaseballteam,assumingthatanyplayercan

playanyposition?

A) 5040ways B) 2520 ways C) 10,080 ways D) 42ways

2) Inhowmanywayscan5volunteersbeassignedto5 boothsforacharitybazaar?

A) 120ways B) 60ways C) 240 ways D) 20ways

3) Arestaurantoffersachoiceof4salads,5 maincourses,and3 desserts.Howmanypossible3–course mealsare

there?

A) 60possiblemeals B) 12possiblemeals C) 20 possiblemeals D) 120 possiblemeals

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4) Lisahas5skirts,10blouses,and4jackets.Howmany3–pieceoutfitscansheputtogetherassuminganypiece

goeswithanyother?

A) 200possibleoutfits B) 19 possibleoutfits

C) 50possibleoutfits D) 400 possibleoutfits

5) Thereare4roadsleadingfromBlufftontoHardeeville,6 roadsleadingfromHardeevilletoSavannah,and3

roadsleadingfromSavannahtoMacon.HowmanywaysaretheretogetfromBlufftontoMacon?

A) 72ways B) 13ways C) 24 ways D) 144 ways

6) Astudentmustchoose1of5scienceelectives,1of6 socialstudieselectives,and1of7languageelectives.How

manypossiblecourseselectionsarethere?

A) 210courseselections B) 18 courseselections

C) 30courseselections D) 420 courseselections

7) Arestaurantoffersachoiceof2salads,9 maincourses,and4 desserts.Howmanypossiblechoicesforameal

arethere(includingsingleitems)?

A) 149possiblemeals B) 134possiblemeals C) 113 possiblemeals D) 87possiblemeals

2 UsethePermutationsFormula

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UsetheformulafornPrtoevaluatetheexpression.

1) 8P3

A) 336 B) 6720 C) 13,440 D) 40,320

2) 5P0

A) 1 B) 50 C) 0 D) 120

Solvetheproblem.

3) Howmany3–lettercodescanbeformedusingthelettersA,B,C,D,andE?Nolettercanbeusedmorethan

once.

A) 60 B) 20 C) 10 D) 40

4) Howmany2–lettercodescanbeformedusingthelettersA,B,C,D,E,andF?Nolettercanbeusedmorethan

once.

A) 30 B) 360 C) 15 D) 720

5) Howmany2–digitnumberscanbeformedusingthedigits1,2,3,4,5,6,7,8,9,and0?Nodigitcanbeused

morethanonce.

A) 90 B) 1,814,400 C) 45 D) 3,628,800

6) InhowmanywayscanSusanarrange10 booksinto3 slotsonherbookshelf?

A) 720 B) 604,800 C) 86,400 D) 1,209,600

7) Thematchingsectionofanexamhas3 questionsand9 possibleanswers.Inhowmanydifferentwayscana

studentanswerthe3questions,ifnoneoftheanswerchoicescanberepeated?

A) 504 B) 60,480 C) 84 D) 120,960

8) Acombinationlockhas30numbersonit.Howmanydifferent3–digitlockcombinationsarepossibleifnodigit

canberepeated?

A) 24,360 B) 8120 C) 4060 D) 870

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9) Achurchhas9bellsinitsbelltower.Beforeeachchurchservice3 bellsarerunginsequence.Nobellisrung

morethanonce.Howmanysequencesarethere?

A) 504 B) 60,480 C) 84 D) 120,960

10) Aclubelectsapresident,vice–president,andsecretary–treasurer.Howmanysetsofofficersarepossibleif

thereare14membersandanymembercanbeelectedtoeachposition?Nopersoncanholdmorethanone

office.

A) 2184 B) 1092 C) 728 D) 24,024

3 DistinguishBetweenPermutationProblemsandCombinationProblems

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Doestheprobleminvolvepermutationsorcombinations?Donotsolve.

1) Thematchingsectionofanexamhas3 questionsand10 possibleanswers.Inhowmanydifferentwayscana

studentanswerthe3questions,ifnoneoftheanswerchoicescanberepeated?

A) permutations B) combinations

2) Inastudentgovernmentelection,5seniors,3 juniors,and2 sophomoresarerunningforelection.Studentselect

fourat–largesenators.Inhowmanywayscanthisbedone?

A) permutations B) combinations

3) Aclubelectsapresident,vice–president,andsecretary–treasurer.Howmanysetsofofficersarepossibleif

thereare9membersandanymembercanbeelectedtoeachposition?Nopersoncanholdmorethanone

office.

A) permutations B) combinations

4) From8namesonaballot,acommitteeof5 willbeelectedtoattendapoliticalnationalconvention.Howmany

differentcommitteesarepossible?

A) permutations B) combinations

4 UsetheCombinationsFormula

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

UsetheformulafornCrtoevaluatetheexpression.

1) 11C3

A) 165 B) 6,652,800 C) 13,305,600 D) 990

2) 5C0

A) 1 B) 50 C) 0 D) 120

Solvetheproblem.

3) Ahamburgershopsellshamburgerswithcheese,relish,lettuce,tomato,onion,mustard,orketchup.How

manydifferenthamburgerscanbeconcoctedusingany4oftheextras?

A) 35 B) 840 C) 210 D) 420

4) Astackof9differentcardsareshuffledandspreadoutfacedown.If3 cardsareturnedfaceup,howmany

different3–cardcombinationsarepossible?

A) 84 B) 504 C) 60,480 D) 252

5) Inastudentgovernmentelection,5seniors,2 juniors,and3 sophomoresarerunningforelection.Studentselect

fourat–largesenators.Inhowmanywayscanthisbedone?

A) 210 B) 5040 C) 151,200 D) 30

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6) From9namesonaballot,acommitteeof3 willbeelectedtoattendapoliticalnationalconvention.Howmany

differentcommitteesarepossible?

A) 84 B) 504 C) 60,480 D) 252

7) Ronfinds9booksatabookstorethathewouldliketobuy,buthecanaffordonly5ofthem.Inhowmany

wayscanhemakehisselection?Howmanywayscanhemakehisselectionifhedecidesthatoneofthebooks

isamust?

A) 126;70 B) 15,120;1680 C) 3024;1680 D) 7560;840

5 AdditionalConcepts

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Evaluatetheexpression.

1) 7P2

2! –7C2

A) 0 B) 2 C) 5040 D) 120

2) 1–4P2

6P4

A) 29

30 B) 1

30 C) 3

5D) 31

30

3) 10C6

8C4

–37!

35!

A) –1329 B) 1329 C) –1403 D) –34

4) 4C2·8C1

18C15

A) 1

17 B) 2

17 C) 1

51 D) 1

11.7 Probability

1 ComputeEmpiricalProbability

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.Roundtothenearesthundredthofapercentifneeded.

1) In1999thestockmarkettookbigswingsupanddown.Asurveyof990 adultinvestorsaskedhowoftenthey

trackedtheirportfolio.Thetableshowstheinvestorresponses.Whatistheprobabilitythatanadultinvestor

trackshisorherportfoliodaily?

Howfrequently? Response

Daily 223

Weekly 283

Monthly 293

Coupletimesayear 139

Donʹttrack 52

A) 22.53% B) 28.59% C) 23.77% D) 29.60%

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2) Useoftheinternetforshoppingisincreasingdramatically,butstillissomewhatagedependent.Whena

popularwebsitethatsellsbooksaskedtheageofuserswhoboughtproductsfromthemovertheinternet,they

obtainedthefollowingdata.Whatistheprobabilitythatabuyeronthiswebsiteisaged60–69?

AgeGroup Number

10–19 1983

20–29 3685

30–39 2864

40–49 654

50–59 386

60–69 311

70–79 105

A) 3.11% B) 3.86% C) 3.91% D) 3.21%

3) Duringclinicaltrialsofanewdrugintendedtoreducetheriskofheartattack,thefollowingdataindicatethe

occurrenceofadversereactionsamong1200adultmaletrialmembers.Whatistheprobabilitythatanadult

maleusingthedrugwillexperiencenausea?

AdverseReaction Number

Heartburn 16

Headache 13

Dizziness 10

Urinaryproblems 7

Nausea 25

Abdominalpain 19

A) 2.08% B) 27.78% C) 1.94% D) 1.58%

4) Measurementsoftheheightofagroupofmenenteringaparticularcollegeproducedthefollowingtable.What

istheprobabilitythatamanenteringthecollegeis68–69inchestall?

Height(inches) 60–61 62–63 64–65 66–67 68–69 70–71 72–73 74–75 76+

Number 5 20 90 249 293 178 81 38 6

A) 30.52% B) 25.94% C) 18.54% D) 30.87%

5) Atrafficengineeriscountingthenumberofvehiclesbytypethatturnintoaresidentialarea.Thetablebelow

showstheresultsofthecountsduringafour–hourperiod.Whatistheprobabilitythatthenextvehiclepassing

isanSUV?

Typeofvehicle Number

Car 265

SUV 416

Van 75

Smalltruck 288

Largetruck 209

Dumptruck 25

Other 73

A) 30.79% B) 19.62% C) 32.55% D) 31.37%

6) DuringJulyinJacksonville,Florida,itisnotuncommontohaveafternoonthunderstorms.Onaverage,8.6 days

haveafternoonthunderstorms.WhatistheprobabilitythatarandomlyselecteddayinJulywillnothavea

thunderstorm?

A) 72.26% B) 71.33% C) 27.74% D) 91.40%

Page27

7) Thetablebelowrepresentsthenumberofdeathsper100casesforanillnesshavingamedianmortalityoffour

yearsandaright–skeweddistributionovertime.Whatistheprobabilityoflivingmorethan12yearsafter

diagnosisofthedisease?

YearsafterDiagnosis Numberdeaths

1–215

3–435

5–616

7–89

9–10 6

11–12 4

13–14 2

15+13

A) 15.00% B) 19.00% C) 13.00% D) 85.00%

2 ComputeTheoreticalProbability

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheprobability.

1) Abagcontains2redmarbles,9bluemarbles,and4 greenmarbles.Whatistheprobabilityofchoosingablue

marblewhenonemarbleisdrawn?

A) 3

5B) 2

15 C) 4

15 D) 9

11

2) A6–sideddieisrolled.Whatistheprobabilityofrollinganumberlessthan2?

A) 1

6B) 1

3C) 5

6D) 1

9

3) Two6–sideddicearerolled.Whatistheprobabilitythesumofthetwonumbersonthediewillbe5?

A) 1

9B) 5

6C) 8

9D) 4

4) Abagcontains7redmarbles,4bluemarbles,and1 greenmarble.Whatistheprobabilityofchoosingamarble

thatisnotbluewhenonemarbleisdrawnfromthebag?

A) 2

3B) 3

2C) 1

3D) 8

5) Abagcontains13ballsnumbered1through13.Whatistheprobabilityofselectingaballthathasaneven

numberwhenoneballisdrawnfromthebag?

A) 6

13 B) 13

6C) 2

13 D) 6

6) Two6–sideddicearerolled.Whatistheprobabilitythatthesumofthetwonumbersonthedicewillbegreater

than9?

A) 1

6B) 1

4C) 1

12 D) 6

7) Alotterygamecontains25ballsnumbered1through25.Whatistheprobabilityofchoosingaballnumbered

26?

A) 0 B) 1 C) 1

25 D) 25

Page28

3 FindtheProbabilitythatanEventWillNotOccur

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheprobability.

1) Whatistheprobabilitythatacarddrawnfromadeckof52cardsisnotred?

A) 1

2B) 2 C) 25

52 D) 3

4

2) Whatistheprobabilitythatacarddrawnfromadeckof52cardsisnota4?

A) 12

13 B) 1

13 C) 9

10 D) 1

10

3) Whatistheprobabilitythatacarddrawnfromadeckof52cardsisnotaclub?

A) 3

4B) 1

4C) 4

13 D) 2

5

4) Abagcontains3bluemarbles,7greenmarbles,and7 redmarbles.Onemarbleisdrawnfromthebag.Whatis

theprobabilitythatthemarbledrawnisnotblue?

A) 14

17 B) 3

14 C) 3

17 D) 14

3

5) Abagcontains24marbles,ofwhich3 areblueand5 aregreen.Onemarbleisdrawnfromthebag.Whatis

theprobabilitythatthemarbledrawnisnotblue?

A) 7

8B) 1

8C) 3

8D) 1

3

4 FindtheProbabilityofOneEventoraSecondEventOccurring

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheprobability.

1) Givetheprobabilitythattherollofadiewillshow4 or3.

A) 1

3B) 2

3C) 1

2D) 2

2) Givetheprobabilitythattherollofadiewillshowanumberlessthan7.

A) 1 B) 7

6C) 1

6

3) Eachoftenticketsismarkedwithadifferentnumberfrom1to10andputinabox.Ifyoudrawaticketfrom

thebox,whatistheprobabilitythatyouwilldraw6,8,or3?

A) 3

10 B) 1

6C) 1

10 D) 1

8

Page29

4)

Whatistheprobabilitythatthearrowwilllandon4or3?

A) 2

5B) 4 C) 1 D) 3

4

5) Whatistheprobabilitythatthearrowwilllandonanoddnumber?

A) 3

5B) 2

5C) 1 D) 0

6) Alotterygamehasballsnumbered1through15.Whatistheprobabilityofselectinganevennumberedballor

a13?

A) 8

15 B) 7

15 C) 2

5D) 7

8

7) Onedigitfromthenumber4,858,778iswrittenoneachofsevencards.Whatistheprobabilityofdrawinga

cardthatshows4or7?

A) 3

7B) 2

7C) 4

7D) 8

7

8) Onedigitfromthenumber3,828,998iswrittenoneachofsevencards.Whatistheprobabilityofdrawinga

cardthatshows3,8,or2?

A) 5

7B) 3

7C) 4

7D) 2

7

9) Acardisdrawnfromawell–shuffleddeckof52cards.Whatistheprobabilityofdrawinganaceora6?

A) 2

13 B) 7

26 C) 13

2D) 7

10) Aspinnerhasregionsnumbered1through15.Whatistheprobabilitythatthespinnerwillstoponaneven

numberoramultipleof3?

A) 2

3B) 7

9C) 1

3D) 12

11) Acardisdrawnfromadeckof52cards.Whatistheprobabilitythatitisanumberedcard(2 –10)ora

diamond?

A) 10

13 B) 53

52 C) 23

52 D) 33

52

12) Acardisdrawnfromadeckof52cards.Whatistheprobabilitythatitisa9 oraheart?

A) 4

13 B) 17

52 C) 2

13 D) 25

26

Page30

13) Acardisdrawnfromadeckof52cards.Whatistheprobabilitythatitisapicturecard(Jack,Queen,King)ora

spade?

A) 11

26 B) 25

52 C) 39

52 D) 7

52

14) Acardisdrawnfromadeckof52cards.Whatistheprobabilitythatitisaspadeorthatitisgreaterthan2 and

lessthan9?

A) 31

52 B) 37

52 C) 7

13 D) 17

26

15) Giventhesequence5

,

6

,

7

,

8…42

,

whatistheprobabilitythatanumberinthesequenceiseven orthatitis

greaterthan8andlessthen25?

A) 27

38 B) 35

38 C) 53

76 D) 53

74

5 FindtheProbabilityofOneEventandaSecondEventOccurring

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheprobability.

1) A6–sideddieisrolled.Whatistheprobabilityofrollinganumberthatisevenanda5?

A) 0 B) 1

6C) 1

2D) 1

2) Iftwo12–sideddicearerolledwhatistheprobabilitythatbothnumberswillbeeven?

A) 1

4B) 1

2C) 71

144 D) 73

144

3) Two6–sideddicearerolled.Whatistheprobabilitythatthesumisoddandthenumberononeofthediceisa

1?

A) 1

6B) 1

36 C) 1

2D) 1

12

4) Acardisdrawnfromawell–shuffleddeckof52cards.Whatistheprobabilityofgettingared4?

A) 1

26 B) 1

52 C) 1

4D) 1

2

5) Acardisdrawnfromawell–shuffleddeckof52cards.Whatistheprobabilitythatthecardwillhaveavalue

of4andbeafacecard?

A) 0 B) 1

13 C) 4

13 D) 3

13

6) UrnAhasballsnumbered1through8.UrnBhasballsnumbered1through4.Whatistheprobabilitythata4

isdrawnfromAfollowedbya2fromB?

A) 1

32 B) 1

16 C) 3

8D) 1

8

7) Anurncontainsballsnumbered1through10.Aballischosen,returnedtotheurn,andasecondballischosen.

Whatistheprobabilitythatthefirstandsecondballswillbea6?

A) 1

100 B) 1

5C) 3

5D) 3

10

Page31

8) Agamespinnerhasregionsthatarenumbered1through9.Ifthespinnerisusedtwice,whatistheprobability

thatthefirstnumberisa3andthesecondisa7?

A) 1

81 B) 1

18 C) 10

81 D) 7

10

Page32

Ch.11 Sequences,Induction,andProbability

AnswerKey

11.1 SequencesandSummationNotation

1 FindParticularTermsofaSequencefromtheGeneralTerm

2 UseRecursionFormulas

3 UseFactorialNotation

4 UseSummationNotation

11.2 ArithmeticSequences

1 FindtheCommonDifferenceforanArithmeticSequence

2 WriteTermsofanArithmeticSequence

3 UsetheFormulafortheGeneralTermofanArithmeticSequence

4 UsetheFormulafortheSumoftheFirstnTermsofanArithmeticSequence

Page34

11.3 GeometricSequencesandSeries

1 FindtheCommonRatioofaGeometricSequence

2 WriteTermsofaGeometricSequence

3 UsetheFormulafortheGeneralTermofaGeometricSequence

Page35

4 UsetheFormulafortheSumoftheFirstnTermsofaGeometricSequence

5 FindtheValueofanAnnuity

6 UsetheFormulafortheSumofanInfiniteGeometricSeries

Page36

11.4 MathematicalInduction

1 UnderstandthePrincipleofMathematicalInduction

Page37

Page38

2 ProveStatementsUsingMathematicalInduction

Page39

Page40

11.5 TheBinomialTheorem

1 EvaluateaBinomialCoefficient

2 ExpandaBinomialRaisedtoaPower

Page41

3 FindaParticularTerminaBinomialExpansion

11.6 CountingPrinciples,Permutations,andCombinations

1 UsetheFundamentalCountingPrinciple

2 UsethePermutationsFormula

3 DistinguishBetweenPermutationProblemsandCombinationProblems

4 UsetheCombinationsFormula

Page42

5 AdditionalConcepts

11.7 Probability

1 ComputeEmpiricalProbability

2 ComputeTheoreticalProbability

3 FindtheProbabilitythatanEventWillNotOccur

4 FindtheProbabilityofOneEventoraSecondEventOccurring

5 FindtheProbabilityofOneEventandaSecondEventOccurring

Page43

Page44