Chapter 9 Geometric Sequences Geometric Series Determine Sequence Geometric

Ch.9 Sequences;Induction;theBinomialTheorem

9.1 Sequences

1 WritetheFirstSeveralTermsofaSequence

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

Evaluatethefactorialexpression.

1) 6!

A) 720 B) 30 C) 12 D) 10

2) 10!

8!

A) 90 B) 2! C) 10

8D) 10

3) 7!

6!

A) 7 B) 1 C) 7

6D) 7!

4) 7!

9!

A) 1

72 B) 72 C) 2! D) 1

2!

5) 2!7!

5!

A) 84 B) 1

60 C) 42 D) 2521

60

6) 4!

2!2!

A) 6 B) 4 C) 0 D) 1

7) 6!

3!3!

A) 20 B) 40 C) 120 D) 10

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

8) {sn}={n–5}

A) s1=–4

,

s2=–3

,

s3=–2

,

s4=–1

,

s5=0B)s

1= –5

,

s2= –4

,

s3=–3

,

s4=–2

,

s5= –1

C) s1=0

,

s2=1

,

s3=2

,

s4=3

,

s5=4D)s

1=5

,

s2=10

,

s3=15

,

s4=20

,

s5=25

9) {sn}={4n–3}

A) s1=1

,

s2=5

,

s3=9

,

s4=13

,

s5=17 B) s1=1

,

s2=2

,

s3=3

,

s4=4

,

s5=5

C) s1=7

,

s2=11

,

s3=15

,

s4=19

,

s5=23 D) s1= –1

,

s2= –5

,

s3=–9

,

s4=–13

,

s5= –17

Page1

10) {sn}={4(2n–3)}

A) s1=–4

,

s2=4

,

s3=12

,

s4=20

,

s5=28 B) s1= –1

,

s2=1

,

s3=3

,

s4=5

,

s5=7

C) s1=–12

,

s2=–4

,

s3=4

,

s4=12

,

s5=20 D) s1= –4

,

s2= –8

,

s3=–12

,

s4=–16

,

s5= –20

11) {sn}={n2–n}

A) s1=0,s2=2,s3=6,s4=12,s5=20 B) s1=2,s2=6,s3=12,s4=20,s5=30

C) s1=1,s2=4,s3=9,s4=16,s5=25 D) s1=0,s2=3,s3=8,s4=15,s5=24

12) {sn}={3n}

A) s1=3

,

s2=9

,

s3=27

,

s4=81

,

s5=243 B) s1=1

,

s2=8

,

s3=27

,

s4=64

,

s5=125

C) s1=1

,

s2=3

,

s3=9

,

s4=27

,

s5=81 D) s1=9

,

s2=27

,

s3=81

,

s4=243

,

s5=729

13) {cn}=4n+4

4n

A) c1=2,c2=3

2,c3=4

3,c4=5

4,c5=6

5B) c1=3

2,c2=4

3,c3=5

4,c4=6

5,c5=7

6

C) c1=2,c2=3,c3=4

3,c4=5,c5=6D)c

1=3

,

c2=4

,

c3=5

,

c4=6

,

c5=7

14) {sn}=2n

3n+2

A) s1=2

5,s2=4

11,s3=8

29,s4=16

83,s5=32

245 B) s1=2

5,s2=1

2,s3=6

11,s4=4

7,s5=10

17

C) s1=4

11,s2=8

29,s3=16

83,s4=32

245 ,s5=64

731 D) s1=1

2,s2=6

11,s3=4

7,s4=10

17,s5=3

5

15) {cn}=(–1)n

(n+5)(n+2)

A) c1=–1

18,c2=1

28,c3=–1

40,c4=1

54,c5=–1

70

B) c1=1

18,c2=–1

28,c3=1

40,c4=–1

54,c5=1

70

C) c1=–1

9,c2=1

11,c3=–1

13,c4=1

15,c5=–1

17

D) c1=1

9,c2=–1

11,c3=1

13,c4=–1

15,c5=1

17

16) {cn}=5n

n

A) c1=5,c2=25

2,c3=125

3,c4=625

4,c5=625 B) c1=0,c2=5,c3=25

2,c4=125

3,c5=625

4

C) c1=5

,

c2=25

,

c3=125

,

c4=625

,

c5=3125 D) c1=1,c2=5

,

c3=25

,

c4=125

,

c5=625

Page2

17) {sn}=(–1)n–1n+2

2n–1

A) s1=3,s2=–4

3,s3=1,s4=–6

7,s5=7

9B) s1=3,s2=4

3,s3=1,s4=6

7,s5=7

9

C) s1=–3,s2=4

3,s3=–1,s4=6

7,s5=–7

9D) s1=–3,s2=4

3,s3=1,s4=–6

7,s5=7

9

18) {sn}=n

n2+2

A) s1=1

3,s2=1

3,s3=3

11,s4=2

9,s5=5

27 B) s1=1

3,s2=1

3,s3=3

8,s4=2

5,s5=5

12

C) s1=1

2,s2=1

3,s3=3

8,s4=2

5,s5=5

12 D) s1=1

4,s2=1

3,s3=3

8,s4=2

5,s5=5

12

Thegivenpatterncontinues.Writedownthenthtermofthesequence{an}suggestedbythepattern.

19) –4

,

4

,

12

,

20

,

28

,

…

A) an=4(2n–3 ) B) an=8n–3C)a

n=–4(8)

n–1D) an=12n–8

20) 1

,

4

,

7

,

10

,

13

,

…

A) an=3n–2B)a

n=2n–3C)a

n=1(3)

n–1D) an=n+3

21) 0,2,6,12,20,…

A) an=n2–nB)a

n=2n –2C)a

n=4n –6D)a

n=2n–1–1

22) 2

,

4

,

8

,

16

,

32

,

…

A) an=2nB) an=2 n C) an=2+2(n–1) D) an=2n–1+1

23) 4

,

–8

,

12

,

–16

,

…

A) an=(–1)n+1·4n B) an=(–1)n·4n C) an=(–1)n+1·4D)a

n=(–1)n·4

24) 1

1·3,1

2·4,1

3·5,1

4·6,…

A) an=1

n(n+2) B) an=1

n·2n C) an=1

2n D) n(n+2)

25) 1,1

3,1

9,1

27,…

A) an=1

3n–1B) an=1

3nC) an=1

3D) an=1

3+n

26) 1

1,1

4,1

9,1

16,1

25,…

A) an=1

n2B) an=1

nn–1C) an=1

3n–2D) an=1

2

n–1

Page3

Solve.

27) Thenumberofstudentsinaschoolinyearnisestimatedbythemodelan=4n2+12n+88.Abouthow

manystudentsareintheschoolineachofthefirstthreeyears?

A) 104

,

128

,

160 B) 116

,

140

,

172 C) 108

,

128

,

160 D) 104

,

128

,

148

28) Duringafive–yearperiod,acompanydoublesitsprofitseachyear.Iftheprofitsattheendofthefifthyear

are$224,000,thenwhataretheprofitsforeachofthefirstfouryears?

A) $14,000

,

$28,000

,

$56,000

,

$112,000 B) $14,000

,

$28,000

,

$56,000

,

$140,000

C) $15,000

,

$30,000

,

$60,000

,

$118,000 D) $14,000

,

$28,000

,

$42,000

,

$56,000

2 WritetheTermsofaSequenceDefinedbyaRecursiveFormula

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

Thesequenceisdefinedrecursively.Writethefirstfourterms.

1) a1=1;an=an–1–5

A) a1=1

,

a2=–4

,

a3=–9

,

a4=–14 B) a1= –5

,

a2= –10

,

a3=–15

,

a4= –20

C) a1=1

,

a2=6,a3=11,a4=16 D) a1=1

,

a2= –2 ,a3=–7,a4=–12

2) a1=6;an=2an–1

A) a1=6

,

a2=12

,

a3=24

,

a4=48 B) a1=6

,

a1=11

,

a2=10

,

a4=9

C) a1=10

,

a1=20

,

a2=40

,

a4=80 D) a1=6

,

a1=14

,

a2=26

,

a4=50

3) a1=3;an=2an–1–1

A) a1=3

,

a2=5

,

a3=9

,

a4=17 B) a1=3

,

a2=5

,

a3=11

,

a4=23

C) a1=3

,

a2=6

,

a3=12

,

a4=24 D) a1=3

,

a2=7

,

a3=15

,

a4=31

4) a1=2,a2=5;an=an–2–3an–1

A) a1=2,a2=5,a3=–13,a4=44 B) a1=2,a2=5,a3=–1,a4=–16

C) a1=2,a2=5,a3=1,a4=2D)a

1=2,a2=5,a3=17,a4=–46

5) a1=135;an+1=1

3

(an)

A) a1=135,a2=45,a3=15,a4=5B)a

1=135,a2=67.5,a3=33.75,a4=16.875

C) a1=135,a2=405,a3=1215,a4=3645 D) a1=135,a2=105,a3=75,a4=45

6) a1=–7;an=n–an–1

A) a1=–7

,

a2=9

,

a3=–6

,

a4=10 B) a1= –7

,

a2= –5

,

a3=8

,

a4=–4

C) a1=–7

,

a2=–14

,

a3=7

,

a4=–21 D) a1= –8

,

a2= –5

,

a3=–7

,

a4=–4

7) a1=–4;an=an–1

n+1

A) a1=–4,a2=–4

3,a3=–1

3,a4=–1

15 B) a1=–4,a2=–2,a3=–2

3,a4=–1

6

C) a1=–4,a2=–4,a3=–4

3,a4=–1

3D) a1=–4,a2=–4,a3=–2,a4=–2

3

Page4

8) a1=w;an=an–1+T

A) a1=w

,

a2=w+T

,

a3=w+2T

,

a4=w+3T B) a1=w

,

a2=T

,

a3=2T

,

a4=3T

C) a1=w

,

a2=w–T

,

a3=w–2T

,

a4=w–3T D) a1=T

,

a2=T+w

,

a3=T+2w

,

a4=T+3w

9) a1=6;an=6an–1

A) a1=6,a2=66,a3=666,a4=6666

B) a1=6,a2=6,a3=66,a4=36

C) a1=6,a2=66,a3=36 6,a4=216 6

D) a1=6,a2=6,a3=6,a4=6

Solve.

10) Giventhata1=–4,a2=–4andan+2=an+1–4an

,

whatisthefifthtermofthisrecursivelydefined

sequence?

A) a5=–20 B) a5=28 C) a5=764 D) a5= –308

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

11) Karenhasabalanceof$1000onadepartmentstorecreditcardthatcharges1.5%interestpermonthon

anyunpaidbalance.Shecanaffordtopay$150towardthebalanceeachmonth.Herbalanceeachmonth,

aftermakinga$150payment,isgivenbytherecursivelydefinedsequence

B0=$1000Bn=1.015Bn–1–150

DetermineKarenʹsbalanceaftermakingthefirstpayment.Thatis,determineB1.

12)

J

akeboughtatruckbytakingoutaloanfor$26,500 at0.25% interestpermonth.Jakeʹsregularmonthly

paymentis$567,buthedecidestopayanextra$75towardthebalanceeachmonth.Hisbalanceeach

month,aftermakinghispayment,isgivenbytherecursivelydefinedsequence

B0=$26,500Bn=1.0025Bn–1–642

DetermineJakeʹsbalanceaftermakingthefirstpayment.Thatis,determineB1.

13) Awildliferefugecurrentlyhas100deerinit.Alocalwildlifesocietydecidestoaddanadditional2deer

eachmonth.Itisalreadyknownthatthedeerpopulationisgrowing12%peryear.Thesizeofthe

populationisgivenbytherecursivelydefinedsequence

p0=100 pn=1.01pn–1+2

Howmanydeerareinthewildliferefugeattheendofthesecondmonth?Thatis,whatisp2?

14) LexingtonReservoirhas300milliongallonsofwater.About0.07%ofthewaterislosttoevaporationevery

week.About150,000gallonsofwaterenterthereservoireveryweek.Theamountofwaterinthereservoir

attheendofeachweekisgivenbytherecursivelydefinedsequence

w0=300 wn=(0.9993)wn–1+0.15

Determinetheamountofwaterinthereservoiratthebeginningofthesecondweek.Thatis,determine

w2.

Page5

15) Mariadeposited$1500inanindependentretirementaccountatherbank.Sheearns0.5%interestper

monthonthebalance.Eachmonth,shedeposits$100intheaccount.Herbalanceeachmonthaftermaking

a$100deposit,isgivenbytherecursivelydefinedsequence

B0=$1500Bn=1.005Bn–1+100

IfshemadetheinitialdepositonSeptember30,andmakeseachmonthlydepositonthelastdayofthe

month,howmuchmoneywillbeintheaccountattheendoftheyearforMariatocountasadeduction

forthatyearʹsfederalincometaxes?Thatis,determineB3.

3 UseSummationNotation

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

Writeoutthesum.

1)

n

k=1

(k+12)

∑

A) 13+14+15+…+(n+12) B) n+12

C) 12+13+14+…+(n+12) D) 1+2+3+…+n

2)

n

k=1

(3k+9)

∑

A) 12+15+18+…+(3n+9) B) 3n+9

C) 9+12+15+…+(3n+9) D) 1+2+3+…+n

3)

n

k=1

(k+2)2

∑

A) 9+16+25+…+(n+2)2B) (n+2)2

C) 3+4+5+…+(n+2)2D) 1+2+3+…+n

4)

n–1

k=0

(4k+3)

∑

A) 3+7+11+…+(4n–1) B) 7 +11 +15 +…+(4n–1)

C) 3+7+11+…+(4n+3) D) 7 +11 +15+…+(4n+3)

5)

n

k=0

3

4

k

∑

A) 1+3

4

+9

16

+…+3

4

nB) 3

4

+9

16

+27

64

+…+3

4

n

C) 0+3

4

+9

16

+…+3

4

nD) 0+1+2+3+…+n

Page6

6)

n–1

k=0

1

4k+1

∑

A) 1

4

+1

16

+1

64

+…+1

4nB) 1+1

4

+1

16

+…+1

4n

C) 0+1

4

+1

16

+…+1

4nD) 1

4

+1

16

+1

64

+…+1

4n+1

7)

n

k=1

k3

4

∑

A) 1

4

+2+27

4

+…+n3

4B) n3

4

C) 4+32+108+…+n3

4D) 1+2+3+…+n

8)

n

k=1

7k+1

∑

A) 72+73+74+…+7n+1B) 7+72+73+…+7n

C) 72+73+74+…+7nD) 7+72+73+…+7n+1

Expressthesumusingsummationnotation.

9) 3+6+9+…+15

A)

5

k=1

3k

∑

B)

5

k=0

3k

∑

C)

5

k=1

k2

∑

D)

5

k=1

3k2

∑

10) 23+33+43+…+83

A)

8

k=2

k3

∑

B)

8

k=1

k3

∑

C)

n

k=2

k3

∑

D)

8

k=3

(k–1)3

∑

11) 3+12+27+...+75

A)

5

k=1

3k2

∑B)

5

k=0

3k2

∑C)

5

k=1

k2

∑D)

5

k=1

32k

∑

12) 1

3

+1

2

+3

5

+…+7

8

A)

14

k=1

k

k+2

∑

B)

14

k=0

k

k+2

∑

C)

n

k=1

k

k+2

∑

D)

14

k=2

k

k+1

∑

Page7

13) 4+42

2

+43

3

+…+4n

n

A)

n

k=1

4k

k

∑B)

n

k=0

4k

k

∑C)

n

k=1

4k

k

∑D)

n

k=0

4k

k

∑

14) 9

e

+18

e2

+27

e3…+9n

en

A)

n

k=1

9k

ek

∑B)

n

k=0

9k

ek

∑C)

n

k=1

9+k

ek

∑D)

n

k=0

9+k

ek

∑

15) 4

5

–16

25

+64

125

–…+(–1)(9+1)4

5

9

A)

9

k=1

(–1)(k+1)

∑4

5

k

B)

10

k=1

(–1)(k+1)

∑4

5

k

C)

n

k=1

(–1)(k+1)

∑4

5

k

D)

9

k=1

4

5

k

∑

16) 49

64

+ 48

65

+47

66

+…+42

611

A)

8

k=1

410–k

63+k

∑B)

4

k=1

410 +k

63+k

∑C)

8

k=1

410 –k

6k

∑D)

8

k=1

4–k

63+k

∑

17) 62+123+184+…+489

A)

8

k=1

(6k)k+1

∑B)

8

k=1

(6k)k

∑

C)

8

k=1

2(k–1)k+1

∑D)

8

k=1

6k2k–1

∑

18) 1

z

+r

2z

+r2

3z

+…+rn–1

nz

A)

n

k=1

rk–1

kz

∑B)

n

k=1

rk

kz

∑C)

n

k=0

rk–1

kz

∑D)

n

k=0

<b>k

kz

∑

Page8

4 FindtheSumofaSequence

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

Findthesumofthesequence.

1)

11

k=1

6

∑

A) 66 B) 6 C) 11 D) 5

2)

8

k=1

(–k)

∑

A) –36 B) –8C)

–7D)

–56

3)

6

k=3

2k

∑

A) 36 B) 12 C) 18 D) 24

4)

5

k=1

(k+8)

∑

A) 55 B) 13 C) 22 D) 42

5)

6

k=3

(4k–4)

∑

A) 56 B) 48 C) 60 D) 32

6)

16

k=1

(2k+7)

∑

A) 384 B) 53 C) 272 D) 400

7)

4

k=1

3k

∑

A) 120 B) 84 C) 39 D) 30

8)

5

k=3

(k2–12)

∑

A) 14 B) –5C)

–12 D) –24

Page9

9)

4

k=2

k(k–9)

∑

A) –52 B) –60 C) –34 D) –9

10)

4

k=1

–1

3

k

∑

A) –20

81 B) 20

81 C) –16

81 D) 40

81

11)

4

k=1

(–1)k·4k

∑

A) 8 B) –40 C) 40 D) 256

9.2 ArithmeticSequences

1 DetermineIfaSequenceIsArithmetic

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

Determinewhetherthesequenceisarithmetic.

1) 4,12,36,108,972,…

A) Arithmetic B) Notarithmetic

2) 2,–3,–8,–13,–18,…

A) Arithmetic B) Notarithmetic

3) 2,4,6,10,12,…

A) Arithmetic B) Notarithmetic

4) 2,–1,–4,–7,–10,…

A) Arithmetic B) Notarithmetic

5) 5,–15,45,–135,405,…

A) Arithmetic B) Notarithmetic

6) 2

,

6

,

10

,

14

,

18

,

…

A) Arithmetic B) Notarithmetic

Anarithmeticsequenceisgiven.Findthecommondifferenceandwriteoutthefirstfourterms.

7) {9–6n}

A) d=–6;3

,

–3

,

–9

,

–15 B) d= –6;–6

,

–12

,

–18

,

–24

C) d=6;3

,

9,15,21 D) d= –6;3

,

–1 ,–7,–13

8) {6n+8}

A) d=6;14

,

20

,

26

,

32 B) d=6;6

,

14

,

20

,

26

C) d=8;14

,

20

,

26

,

32 D) d=8;6

,

14

,

20

,

26

Page10

9) 1

6

+n

7

A) d=1

7;13

42,19

42,25

42,31

42 B) d=1

6;1

6,13

42,19

42,25

42

C) d=1

7;1

6,13

42,19

42,25

42 D) d=1

6;13

42,19

42,25

42,31

42

2 FindaFormulaforanArithmeticSequence

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

Findthenthtermandtheindicatedtermofthearithmeticsequencewhoseinitialterm,a,andcommondifference,

d,aregiven.

1) a=8;d=2

an=?;a16=?

A) an=6+2n;a16=38 B) an=8+2n;a16 =38

C) an=6+2n;a16=26 D) an=6–2n;a16 =38

2) a=89;d=–8

an=?;a10=?

A) an=97–8n;a10=17 B) an=89 –8n;a10=17

C) an=97–8n;a10=–7D)a

n=89 –8n;a10=–7

3) a=6;d=–6

an=?;a15=?

A) an=12–6n;a15=–78 B) an=6–6n;a15 =–78

C) an=12–6n;a15=–36 D) an=12 +6n;a15=–78

4) a=–9;d=6

an=?;a9=?

A) an=–15+6n;a9=39 B) an= –9+6n;a9=39

C) an=–15+6n;a9=69 D) an= –15 –6n;a9=39

Findtheindicatedtermofthesequence.

5) Thetenthtermofthearithmeticsequence8

,

14

,

20

,

…

A) 62 B) 68 C) –46 D) –52

6) Thenineteent

h

termofthearithmeticsequence0,8

,

16

,

…

A) 144 B) 152 C) 160 D) 126

7) Thethirteent

h

termofthearithmeticsequence29

,

25

,

21

,

…

A) –19 B) –23 C) –48 D) 77

8) Thetwentiethtermofthearithmeticsequence–10 3,–63,–23,…

A) 66 3 B) 70 3 C) –86 3 D) –90 3

Findthefirstterm,thecommondifference,andgivearecursiveformulaforthearithmeticsequence.

9) 7thtermis78;14thtermis57

A) a1=96

,

d=–3

,

an=an–1–3B)a

1=96

,

d=3

,

an=an–1+3

C) a1=99

,

d=–3

,

an=an–1–3D)a

1=99

,

d=3

,

an=an–1+3

Page11

10) 10thtermis–38;13thtermis–53

A) a1=7

,

d=–5

,

an=an–1–5B)a

1=7

,

d=5

,

an=an–1+5

C) a1=12

,

d=–5

,

an=an–1–5D)a

1=12

,

d=5

,

an=an–1+5

11) 7thtermis45;13thtermis93

A) a1=–3

,

d=8

,

an=an–1+8B)a

1= –3

,

d=8

,

an=an–1–8

C) a1=–11

,

d=8

,

an=an–1+8D)a

1= –11

,

d=8

,

an=an–1–8

12) 6thtermis–10;15thtermis–46

A) a1=10

,

d=–4,an=an–1–4B)a

1= –30

,

d=4,an=an–1+4

C) a1=–10

,

d=–4,an=an–1–4D)a

1=10

,

d=4,an=an–1+4

Findtheindicatedtermusingthegiveninformation.

13) a=6

,

d=5;a8

A) 41 B) 46 C) –29 D) –34

14) a=–1

,

d=–4;a25

A) –97 B) –101 C) 95 D) 99

15) a=–1,d=–1

2;a31

A) –16 B) –33

2C) 14 D) 29

2

16) –6

,

–1

,

4

,

…;a36

A) 169 B) 174 C) –181 D) –186

17) a 17 =47

,

a12=37;a1

A) 15 B) 17 C) 2 D) 19

18) a 15 =59

,

a12=44;a7

A) 19 B) 24 C) 5 D) –11

19) a 49 =–151

2,a12=–20;a5

A) –19

2B) –11 C) 7 D) –3

2

Solvetheproblem.

20) Thepopulationofatownisincreasingby300 inhabitantseachyear.Ifitspopulationatthebeginningof

1990was27,541,whatwasitspopulationatthebeginningof1995?

A) 29,041inhabitants B) 137,675 inhabitants

C) 28,741inhabitants D) 275,350 inhabitants

3 FindtheSumofanArithmeticSequence

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

Findthesum.

1) (–6)+(–1)+4+9+…+39

A) 165 B) 170 C) 160 D) 330

Page12

2) 1+2+3+…+295

A) 43660 B) 43365 C) 87,320 D) 87,025

3) 2+4+6+…+336

A) 28,392 B) 28,224 C) 28,056 D) 28,561

4) 1+3+5+…+1059

A) 280,900 B) 281,430 C) 281,961 D) 279,841

5) 6+12+18+…+624

A) 32760 B) 32448 C) 33075 D) 32136

6) –3+1+5+9+13+…+(4n–7)

A) n(2n–5) B) n(4n–7) C) n(2n+5) D) n(4n+7)

7)

47

n=1

(2n+3)

∑

A) 2397 B) 2350 C) 2585 D) 2749.5

8)

33

n=1

(6n–4)

∑

A) 3234 B) 3135 C) 3399 D) 3514.5

9)

26

n=1

(–2n+5)

∑

A) –572 B) –546 C) –442 D) –351

10)

48

n=1

(–4n–4)

∑

A) –4896 B) –4800 C) –4728 D) –4560

11)

10

n=1

(3.6n+8.83)

∑

A) 286.3 B) 44.83 C) 41.23 D) 241.47

Solve.

12) Atheaterhas34rowswith24seatsinthefirstrow,27 inthesecondrow,30 inthethirdrow,andsoforth.

Howmanyseatsareinthetheater?

A) 2499seats B) 2550 seats C) 4998 seats D) 5100 seats

13) Abrickstaircasehasatotalof19stepsThebottomsteprequires146 bricks.Eachsuccessivesteprequires5

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

A) 1919bricks B) 3629 bricks C) 3838 bricks D) 1871.5 bricks

14) Supposeyoujustreceivedajobofferwithastartingsalaryof$37,000 peryearandaguaranteedraiseof

$1500peryear.Howmanyyearswillitbebeforeyouʹvemadeatotal(oraggregate)salaryof$1,025,000?

A) 20years B) 21years C) 25years D) 18years

Page13

15) Alocalcivictheaterhas22seatsinthefirstrowand21rowsinall.Eachsuccessiverowcontains3

additionalseats.Howmanyseatsareinthecivictheater?

A) 1092seats B) 790seats C) 1070seats D) 1010seats

9.3 GeometricSequences;GeometricSeries

1 DetermineIfaSequenceIsGeometric

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

Ageometricsequenceisgiven.Findthecommonratioandwriteoutthefirstfourterms.

1) {sn}=41

2

n–1

A) an=41

2

n–1

r=1

2;s1=4,s2=2,s3=1,s4=1

2

B) an=4·(2)n–1

r=2;s1=4,s2=8,s3=16,s4=32

C) an=41

4

n–1

r=1

4;s1=4,s2=1,s3=1

4,s4=1

16

D) an=1

2

(4)n–1

r=1

2;s1=4,s2=8,s3=16,s4=32

2) {sn}={2n}

A) r=2;s1=2

,

s2=4

,

s3=8

,

s4=16 B) r=2;s1=2

,

s2=4

,

s3=6

,

s4=8

C) r=2n;s1=2

,

s2=4

,

s3=8

,

s4=16 D) r=2n;s1=2

,

s2=4

,

s3=6

,

s4=8

3) {tn}=3

2

n

A) r=3

2;t1=3

2,t2=9

4,t3=27

8,t4= 81

16 B) r=3

2n;t1=3

2,t2=9

4,t3=27

8,t4=81

16

C) r=3

2;t1=3

2,t2=3

2,t3=3

2,t4=3

2D) r=3

2n;t1=3

2,t2=3

2,t3=3

2,t4=3

2

4) {dn}=3n

12

A) r=3;d1=1

4,d2=3

4,d3=9

4,d4=27

4B) r=3

4;d1=1

4,d2=3

4,d3=9

4,d4=27

4

C) r=3;d1=3

4,d2=9

4,d3=27

4,d4=81

4D) r=3

4;d1=3

4,d2=9

4,d3=27

4,d4=81

4

5) {sn}={43n}

A) r=64;s1=64

,

s2=4096

,

s3=262,144

,

s4=16,777,216

B) r=4;s1=64

,

s2=4096

,

s3=262,144

,

s4=16,777,216

C) r=12;s1=12

,

s2=24

,

s3=36

,

s4=48

D) r=64;s1=12

,

s2=24

,

s3=36

,

s4=48

Page14

6) {un}=4n

5n–1

A) r=4

5;u1=4,u2=16

5,u3=64

25,u4=256

125 B) r=4

5;u1=4

5,u2=16

25,u3=64

125 ,u4=256

625

C) r=4;u1=4,u2=16

5,u3=64

25,u4=256

125 D) r=4;u1=4,u2=16

5,u3=64

5,u4=256

5

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.

7) {3n–4}

A) Arithmetic;d=3 B) Geometric;r=3 C) Arithmetic;d= –4 D) Neither

8) {3n2}

A) Geometric;r=3

2B) Arithmetic;d=3 C) Geometric;r=3 D) Neither

9) 4

3

n

A) Geometric;r=4

3B) Arithmetic;d=4

3C) Geometric;r=3

4D) Neither

10) {4n2–2}

A) Arithmetic;d=–2 B) Geometric;r=4 C) Arithmetic;d=4 D) Neither

11) 5–3

4n

A) Arithmetic;d=–3

4B) Geometric;r=–3

4

C) Arithmetic;d=5 D) Neither

12) {4n

/

5}

A) Geometric;r=41

/

5B) Arithmetic;d=41

/

5

C) Geometric;r=4 D) Neither

13) 3

,

5

,

8

,

12

,

…

A) Geometric;r=3 B) Arithmetic;d=3 C) Geometric;r=9 D) Neither

14) 3

,

–9

,

27

,

–81

,

243

,

…

A) Geometric;r=–3 B) Geometric;r=3

C) Arithmetic;d=–12 D) Neither

15) 3,5,7,11,13,…

A) Geometric;r=2 B) Arithmetic;d=2 C) Arithmetic;d=4 D) Neither

Page15

2 FindaFormulaforaGeometricSequence

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

Findthefifthtermandthenthtermofthegeometricsequencewhoseinitialterm,a,andcommonratio,r,aregiven.

1) a=8;r=5

A) a5=5000;an=8·(5)n–1B) a5=5000;an=8·(5)n

C) a5=25,000;an=8·(5)nD) a5=25,000;an=8·(5)n–1

2) a=5;r=1

4

A) a5=5

256 ;an=5·1

4

n–1B) a5=5

256 ;an=5·1

4

n

C) a5=5

1024;an=5·1

4

n–1D) a5=5

1024;an=5·1

4

n

3) a=5;r=–3

A) a5=405;an=5·(–3)n–1B) a5=405;an=5·(–3)n

C) a5=–135;an=5·(–3)n–1D) a5=–135;an=5·(–3)n

4) a=–5;r=–3

A) a5=–405;an=–5·(–3)n–1B) a5=–405;an=–5·(–3)n

C) a5=135;an=–5·(–3)n–1D) a5=135;an=–5·(–3)n

5) a=2;r=5π

A) a5=1250π4,an=2·5n–1πn–1B) a5=6250π5,an=2·5nπn

C) a5=1250π,an=2·5n–1πD) a5=2+20π

,

an=2+5π(n–1)

6) a=10;r=10

A) a5=100 10,an=10n/2 B) a5=100,000,an=10n

C) a5=10,000 10,an=10n–1/2 D) a5=10 10,an=10n/2–1

Findtheindicatedtermofthegeometricsequence.

7) 7thtermof1,1

2,1

4,…

A) a7=1

64 B) a7=1

2C) a7=1

32 D) a7=1

256

8) 7thtermof1,3

,

9

,

…

A) a7=729 B) a7=6561 C) a7=2187 D) a7=9

9) 6thtermof–1,2

,

–4

,

…

A) a6=32 B) a6= –32 C) a6= –64 D) a6=64

10) 7thtermof0.7

,

0.07

,

0.007

,

...

A) 0.0000007 B) 0.00000007 C) 0.000007 D) 0.00000001

Page16

Findthenthterm{an}ofthegeometricsequence.Whengiven,risthecommonratio.

11) a=2;r=–5

A) an=2·(–5)n–1B) an=2·(–5)nC) an=2·5nD) an=–5·(2)n–1

12) a=–3;r=1

3

A) an=–31

3

n–1B) an=–31

3

nC) an=–3–1

3

n–1D) an=1

3

(–3)n–1

13) 4,2,1,1

2,…

A) an=41

2

n–1B) an=4·2n–1C) an=41

4

n–1D) an=1

2

(4)n–1

14) 2

,

6

,

18

,

54

,

162

,

…

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

15) –2

,

–6

,

–18

,

–54

,

–162

,

…

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

16) 2

,

–4

,

8

,

–16

,

32

,

…

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

17) 3,3

5,3

25,3

125 ,3

625 ,…

A) an=3·1

5

n–1B) an=3·1

5

nC) an=3·1

5

n+1D) an=3·1

25

n–1

18) 4,–4

5,4

25,–4

125 ,4

625 ,…

A) an=4·–1

5

n–1B) an=4·–1

5

nC) an=4·–1

5

n+1D) an=4·–1

25

n–1

Solve.

19) Findthe10thtermofthegeometricsequence1

3,1,3,…

A) 6561 B) 19683 C) 2187 D) 177,147

20) Forthegeometricsequence2,1,1

2,1

4,…,findan.

A) an=1

2

1–nB) an=1

2

n–2C) an=21–nD) an=2n–2

21) Forthegeometricsequence64,16,4,1,…,findan.

A) an=1

4

n–4B) an=1

4

nC) an=4n–4D) an=4n

Page17

22) Anewpieceofequipmentcostacompany$46,000.Eachyear,fortaxpurposes,thecompanydepreciates

thevalueby25%.Whatvalueshouldthecompanygivetheequipmentafter7years?

A) $6140 B) $3 C) $8187 D) $11

23) Aparticularsubstancedecaysinsuchawaythatitloseshalfitsweighteachday.Howmuchofthe

substanceisleftafter7daysifitstartsoutat32grams?

A) 1

4

gram B) 4grams C) 1

2

gram D) 2grams

24) Abicyclewheelrotates500timesinaminuteaslongastheriderispedaling.Iftheriderstopspedaling,

thewheelstartstoslowdown.Eachminuteitwillrotateonly2/3asmanytimesasinthepreceding

minute.Howmanytimeswillthewheelrotateinthe4thminuteaftertheriderʹsfeetleavethepedals?

Roundyouranswertothenearestunit.

A) 99times B) 148times C) 0times D) 19times

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

only71%asfarasonthepreviousswing.Whatisthelengthofthearcafter7swings?Roundyouranswer

totwodecimalplaces,ifnecessary.

A) 5.12inches B) 3.64 inches C) 2.58 inches D) 170.4 inches

26) Abaseballplayersignsacontractwithastartingsalaryof$880,000 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sixthyear?

A) $1,096,640 B) $1,097,565 C) $1,095,388 D) $1,099,087

27) Keyanatakesajobwithastartingsalaryof$33,000 forthefirstyearwithanannualincreaseof6.5%

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

A) $48,152 B) $49,014 C) $46,791 D) $50,604

3 FindtheSumofaGeometricSequence

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

Findthesum.

1) 1

7

+3

7

+32

7

+33

7

+…+3n–1

7

A) –1

14

(1–3n)B)

–1

7

(1–3n)C)

–2

7

(1–3n)D)

1

14

(1–3n)

2)

5

k=1

1

3(4)k

∑

A) 1364

3B) 1394

3C) 1388

3D) 1304

3

3)

4

k=1

3

5

k+1

∑

A) 2448

3125 B) 816

625 C) 2452

3125 D) 12969

3125

Page18

4)

n

k=1

5·7k–1

∑

A) –5

6

(1–7n)B)

–5

6

(1–7n–1)C)

–30(1–7n)D)

–5(1–7n)

Useagraphingutilitytofindthesumofthegeometricsequence.Roundanswertotwodecimalplaces,ifnecessary.

5) 3+9+27+81+243+…+3·38

A) 29,523 B) 29,503 C) 29,560 D) 29,525

6) –2–4–8–16–32–…–2·212

A) –16,382 B) –16,402 C) –16,345 D) –16,380

7) 5–15+45–135+405–…+5·(–3)9

A) –73,810 B) –73,812 C) –73,803 D) –73,816

8) 1

5

+2

5

+22

5

+23

5

+…+213

5

A) 3276.6 B) 3276.4 C) 6553.2 D) 6553.4

9)

5

k=1

3(3)k

∑

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

10)

5

k=1

4(–2)k

∑

A) –88 B) –340 C) 8 D) 40

11)

13

k=1

1

3

·(–3)k–1

∑

A) 132,860.33 B) 132,859.67 C) 132,862.67 D) 132,858.33

Solve.

12) Asmallbusinessownermade$50,000 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) $54,636.35;$209,181.35 B) $1.35;$51,546.35

C) $109,850.00;$309,350.00 D) $142,805.00;$452,155.00

13) 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70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) 46min;287min B) 41min;287 min C) 46 min;241 min D) 41min;241 min

Page19

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

14) Initially,apendulumswingsthroughanarcof3feet.Oneachsuccessiveswing,thelengthofthearcis0.8

ofthepreviouslength.After10swings,whattotallengthwillthependulumhaveswung(tothenearest

tenthofafoot)?

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

15)

J

oytownhasapresentpopulationof40,000andthepopulationisincreasingby2.5%eachyear.Howlong

willittakeforthepopulationtodouble?Roundyouranswertothenearestyear.

A) 28years B) 40years C) 29years D) 41years

4 DetermineWhetheraGeometricSeriesConvergesorDiverges

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

Determinewhethertheinfinitegeometricseriesconvergesordiverges.Ifitconverges,finditssum.

1) 1+1

5

+1

25

+1

125

+...

A) Converges;5

4B) Converges;6

5C) Converges;1

5D) Diverges

2) 2–1

2

+1

8

–1

32

+...

A) Converges;8

5B) Converges;3

2C) Converges;–1

2D) Diverges

3) 80+20+5+5

4

+...

A) Converges;320

3B) Converges;–80

3C) Converges;105 D) Diverges

4) 3+3

2

+3

4

+…

A) Converges;6 B) Converges;3 C) Converges;3

2D) Diverges

5) 1–1

3

+1

9

–…

A) Converges;3

4B) Converges;1 C) Converges;–1

3D) Diverges

6) 20+10+5+…

A) Converges;40 B) Converges;–20 C) Converges;35 D) Diverges

7) –15–3–3

5

–…

A) Converges;–75

4B) Converges;15

4C) Converges;–93

5D) Diverges

8) 15+19+21+23+…

A) Converges;34 B) Converges;45 C) Converges;∞D) Diverges

Page20

9)

∞

k=1

1

9

·3k–1

∑

A) Converges;1

3B) Converges;10

3C) Converges;∞D) Diverges

10)

∞

k=1

42

3

k–1

∑

A) Converges;12 B) Converges;16 C) Converges;4 D) Diverges

Solve.

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

only90%asfarasonthepreviousswing.Howfarwillitswingaltogetherbeforecomingtoacomplete

stop?

A) 400inches B) 44inches C) 200 inches D) 89inches

12) Aballisdroppedfromaheightof25feet.Eachtimeitstrikestheground,itbouncesupto0.7ofthe

previousheight.Thetotaldistancetheballhastraveledbeforethesecondbounceis25+2(25·0.7)feet,

andthetotaldistancetheballhastraveledbeforebouncen+1is

25+

n

k=1

500.7k

∑feet.

Usefactsaboutinfinitegeometricseriestocalculatethetotaldistancetheballhastraveledbythetimeit

hasstoppedbouncing.

A) 141 2

3

feet B) 141 2

5

feet C) 140 1

3

feet D) 144 3

5

feet

13) Aping–pongballisdroppedfromaheightof9ftandalwaysrebounds1

3

ofthedistancefallen.Findthe

totalsumofthereboundheightsoftheball.

A) 4.5ft B) 13.5ft C) 6ft D) 3ft

14) Afterbeingstruckwithahammer,agongvibrates55 vibrationsinthefirstsecondandineachsecond

thereaftermakes5

6

asmanyvibrationsasintheprevioussecond.Findhowmanyvibrationsthegong

makesbeforeitstopsvibrating.

A) 330vibrations B) 66vibrations C) 340 vibrations D) 61vibrations

Page21

5 SolveAnnuityProblems

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

Solve.

1)

J

ackdecidedtoput$600intoanIRAaccountevery3monthsatarateof6%compoundedquarterly.Finda

recursiveformulathatrepresentshisbalanceattheendofeachquarter.Howlongwillitbebeforethe

valueoftheaccountis$100,000?Whatwillbethebalancein30yearswhenJackretires?

A) B0=600Bn=(1+0.06

4)B(n–1)+600;21years;$202,355

B) B0=600Bn=(1+0.06

3)B(n–1)+600;18years;$151,860

C) B0=600Bn=(1+0.6

4)B(n–1)+600;7years;$4million

D) B0=600Bn=(1+0.06

12 )B(n–1)+600;10years;$6.8million

2) Afterworkingfor25yearsyouwouldliketohave$800,000 inanannuityforearlyretirement.Ifthe

annualinterestrateis7.5%,compoundedmonthly,whatwillyourmonthlydepositneedtobe?

A) $911.93 B) $2666.67 C) $3088.48 D) $1146.98

3) Tosaveforretirement,youdecidetodeposit$2500 intoanIRAattheendofeachyearforthenext35

years.Iftheinterestrateis8%peryearcompoundedannually,findthevalueoftheIRAafter35years.

(Roundtothenearestdollar.)

A) $430,792 B) $34,463 C) $396,567 D) $11,982,075

4) Lookingaheadtoretirement,yousignupforautomaticsavingsinafixed–income401Kplanthatpays5%

peryearcompoundedannually.Youplantoinvest$3500attheendofeachyearforthenext15years.

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

A) $75,525 B) $77,295 C) $73,982 D) $76,823

5) Lonniedeposits$200eachmonthintoanaccountpayingannualinterestof7%compoundedmonthly.

Howmuchwillhisaccounthaveinitattheendof6years?

A) $17,832 B) $1431 C) $17,678 D) $17,961

6) Laurainvests$250eachquarterinafixed–interestmutualfundpayingannualinterestof5%compounded

quarterly.Howmuchwillheraccounthaveinitattheendof8years?

A) $9763 B) $2387 C) $29,281 D) $9892

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

7) Youdeposit$150each6monthsintoanannuitywithanannualinterestrateof13.5%

,

compounded

semiannually.

a) Whatisthebalanceafter20years?

b) Whatisthebalanceafter40years?

8) Christinecontributes$100eachmonthtoher401(k).Tothenearestdollar,whatwillbethevalueof

Christineʹs401(k)in20yearsiftheperannumrateofreturnisassumedtobe10%compoundedmonthly.

Page22

9.4 MathematicalInduction

1 ProveStatementsUsingMathematicalInduction

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

UsethePrincipleofMathematicalInductiontoshowthatthestatementistrueforallnaturalnumbersn.

1) 4+9+14+…+(5n–1)=n

2(5n+3)

2) 7+14+21+…+7n=7n(n+1)

2

3) 1+3+32+…+3n–1=3n–1

2

4) n2–n+2isdivisibleby2

5) 1

2

+1

4

+1

8

+1

16

+…+1

2n

=1–1

2n

6) UsethePrincipleofMathematicalInductiontoshowthatthestatementʺ5isafactorof7n–2nʺistruefor

allnaturalnumbers.(Hint:7k+1–2k+1=7(7k–2k)+5·2k)

7) Showthattheformula

2+4+6+8+…+2n=n2+n+3

obeysConditionIIofthePrincipleofMathematicalInduction.Thatis,showthatiftheformulaistruefor

somenaturalnumberk,itisalsotrueforthenextnaturalnumberk+1.Thenshowthattheformulais

falseforn=1.

8) Showthatthestatementʺn2–n+3isaprimenumberʺistrueforn=1,butisnottrueforn=3.

9) 6+12+18+…+6n=3n(n+1)

10) 5+5·1

3

+5·1

3

2+…+5·1

3

n–1=

51–1

3

n

1–1

3

11) 12+42+72+...+(3n–2)2=n(6n2–3n–1)

2

12) 1·2+2·3+3·4+...+n(n+1)=n(n+1)(n+2)

3

13) 1–1

2

1–1

3

...1–1

n+1

=1

n+1

14) 1·6+2·6+3·6+...+6n=6n(n+1)

2

Page23

15) 25n=25n

9.5 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expression.

1)

12

8

A) 495 B) 11,880 C) 247 D) 1

2) 5

2

A) 10 B) 5 C) 0 D) 1

3) 8

4

A) 70 B) 140 C) 1680 D) 35

4)

214

2

A) 22,791 B) 45,582 C) 214!

212! D) 212

5)

7

0

A) 1 B) 5040 C) 2 D) 0

6) 6

1

A) 6 B) 1 C) 6

5D) 6!

7)

10

A) 1 B) 3,628,800 C) 2 D) 0

2 UsetheBinomialTheorem

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

ExpandtheexpressionusingtheBinomialTheorem.

1) (x+1)6

A) x6+6x5+15x4+20x3+15x2+6x+1B)x

6+6x5+15x4+20x3+15x2+6x+6

C) x6+6x5+30x4+120x3+30x2+6x+1D)x

6+6x5+30x4+120x3+360x2+720x+720

2) (x–2)5

A) x5–10x4+40x3–80x2+80x–32 B) x5–10x4+40x3–80x2+80x–2

C) x5–10x4+80x3–160x2+80x–32 D) x5–10x4+80x3–160x2+80x–2

Page24

3) (3x+4)3

A) 27x3+108x2+144x+64 B) 27x3+108x2+108x+64

C) 9x6+12x3+4096 D) 9x2+24x+16

4) (3x+2)4

A) 81x4+216x3+216x2+96x+16 B) 162x4+432x3+216x2+192x+16

C) (9x2+6x+4)4D) 81x3+216x2+216x+96

5) (3x–1)5

A) 243x5–405x4+270x3–90x2+15x–1 B) 243x5–81x4+27x3–9x2+3x–1

C) (9x2–6x+1)5D) 243x5+15x4–90x3–90x2+15x–1

6) (3x2+1)3

A) 27x6+27x4+9x2+1B)(9x

4+6x2+1)3

C) 27x3+27x2+9x+1 D) 81x8+27x6+27x4+9x2+1

7) (2x–2y)3

A) 8x3–24x2y+24xy2–8y3B) 8x3–8x2y+8xy2–8y3

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

8) (x2+4y)4

A) x8+16x6y+96x4y2+256x2y3+256y4B) x8+4x6y+96x4y2+128x2y3+256y4

C) x4+16x3y+96x2y2+256xy3+256y4D) x8+16x6y+96x4y2+16x2y3+256y4

9) (x5+5y)4

A) x20+20x15y+150x10y2+500x5y3+625y4B) x9+15x8y+150x7y2+375x5y3+625y4

C) x9+20x8y+30x7y2+20x5y3+5y4D) x20+15x15y+150x10y2+375x5y3+625y4

10) (2x+y)6

A) 64x6+192x5y+240x4y2+160x3y3+60x2y4+12xy5+y6

B) 2x6+12x5y+30x4y2+40x3y3+12xy5+y6

C) 64x6+192x5y+480x4y2+960x3y3+1440x2y4+12xy5+y6

D) 64x6+192x5y+240x4y2+160x3y3+240x2y4+192xy5+64y6

11) (w–s)6

A) w6–6w5s+15w4s2–20w3s3+15w2s4–6ws5+s6

B) w6–s6

C) w6–8w5s+17w4s2–22w3s3+17w2s4–8ws5+s6

D) w6–6w5s–30w4s2+120w3s3+360w2s4–720ws5–720s6

12) (g–2h)3

A) g3–6g2h+12gh2–8h3B) g3–3g2h+6gh2–2h3

C) g3–6h2g+12hg2–8h3D) g3–8h3

Page25

13) x–3

x

4

A) x4–12x5/2+54x–108

x

+81

x2B) x4–12x3/2+54x–108

x

+81

x2

C) x4–81

x2D) x4+81

x2

14) ( x+7)4

A) x2+47

x3/2+42x+28 7x1/2+49 B) x2+27x3/2+21x+14 7x1/2+49

C) x2+47

x1/2+42x+28 7x1/2+2401 D) x2+49

15) (qx+sy)5

A) q5x5+5q4sx4y+10q3s2x3y2+10q2s3x2y3+5qs4xy4+s5y5

B) q5x5+5q3s2x3y2+5qs4xy4+s5y5

C) q5x5+5qsx4y+10q3s2x3y2+10q2s3x2y3+5qsxy4+s5y5

D) q5x5+s5y5

UsetheBinomialTheoremtofindtheindicatedcoefficientorterm.

16) Thecoefficientofxintheexpansionof(9x+6)3

A) 972 B) 1458 C) 36 D) 1944

17) Thecoefficientofxintheexpansionof(4x+3)5

A) 1620 B) 2160 C) 1215 D) 540

18) Thecoefficientof1

x

intheexpansionof 2x+1

x

3

A) 6 B) 2 C) 9 D) 3

19) Thecoefficientofx4intheexpansionof(3x+4)6

A) 19,440 B) 34,560 C) 11,664 D) 23,328

20) Thecoefficientofx8intheexpansionof(x2–3)7

A) –945 B) 945 C) 2835 D) –2835

21) The3rdtermintheexpansionof(2x+4)3

A) 96x B) 48x2C) 16 D) 192x

22) The5thtermintheexpansionof(2x+4)5

A) 2560x B) 1280x2C) 5120 D) 640x

23) The4thtermintheexpansionof(4x–2y)10

A) –15,728,640x7y3B) 7,864,320x7y4C) 7,864,320x3y7D) –3,932,160x3y7

24) The9thtermintheexpansionof(x+2y)13

A) 329,472x5y8B) 164,736x5y9C) 164,736x8y5D) 329,472x8y5

Page26

25) The5thtermintheexpansionof(4x+3y)9

A) 10,450,944x5y4B) 3,483,648x5y5C) 3,483,648x4y5D) 2,612,736x4y5

26) The5thtermintheexpansionof(x+2y)9

A) 2016x5y4B) 1008x5y5C) 1008x4y5D) 2016x4y5

Solve.

27) UsetheBinomialTheoremtoapproximate(1.01)5=(1+10–2)5to7decimalplaces.

A) 1.0510101 B) 1.050101 C) 1.0550101 D) 1.0550505

28) UsetheBinomialTheoremtoapproximate(0.98)6=(1–2(10–2))6to5decimalplaces.

A) 0.88584 B) 0.85841 C) 0.88854 D) 0.88548

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Ch.9 Sequences;Induction;theBinomialTheorem

AnswerKey

9.1 Sequences

1 WritetheFirstSeveralTermsofaSequence

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