Elementary Education Chapter 13 The number of students in a school in year n is estimated by the model

Ch.13 Sequences;Induction;theBinomialTheorem

13.1 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) 8!

A) 40,320 B) 56 C) 16 D) 12

2) 4!

2!

A) 12 B) 2! C) 4

2D) 4

3) 7!

6!

A) 7 B) 1 C) 7

6D) 7!

4) 6!

8!

A) 1

56 B) 56 C) 2! D) 1

2!

5) 4!9!

4!

A) 362,880 B) 1 C) 15120 D) 15121

6) 8!

6!2!

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

7) 12!

6!6!

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

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

8) {sn}={n–4}

A) s1=–3

,

s2=–2

,

s3=–1

,

s4=0

,

s5=1B)s

1= –4

,

s2= –3

,

s3=–2

,

s4=–1

,

s5=0

C) s1=–1

,

s2=0

,

s3=1

,

s4=2

,

s5=3D)s

1=4

,

s2=8

,

s3=12

,

s4=16

,

s5=20

9) {sn}={2n–2}

A) s1=0

,

s2=2

,

s3=4

,

s4=6

,

s5=8B)s

1=0

,

s2=1

,

s3=2

,

s4=3

,

s5=4

C) s1=4

,

s2=6

,

s3=8

,

s4=10

,

s5=12 D) s1=0

,

s2= –2

,

s3=–4

,

s4=–6

,

s5= –8

10) {sn}={2(2n–3)}

A) s1=–2

,

s2=2

,

s3=6

,

s4=10

,

s5=14 B) s1= –1

,

s2=1

,

s3=3

,

s4=5

,

s5=7

C) s1=–6

,

s2=–2

,

s3=2

,

s4=6

,

s5=10 D) s1= –2

,

s2= –4

,

s3=–6

,

s4=–8

,

s5= –10

Page1

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}={2n}

A) s1=2

,

s2=4

,

s3=8

,

s4=16

,

s5=32 B) s1=1

,

s2=4

,

s3=9

,

s4=16

,

s5=25

C) s1=1

,

s2=2

,

s3=4

,

s4=8

,

s5=16 D) s1=4

,

s2=8

,

s3=16

,

s4=32

,

s5=64

13) {cn}=2n+4

2n

A) c1=3,c2=2,c3=5

3,c4=3

2,c5=7

5B) c1=2,c2=5

3,c3=3

2,c4=7

5,c5=4

3

C) c1=3,c2=4,c3=5

4,c4=6,c5=7D)c

1=4

,

c2=5

,

c3=6

,

c4=7

,

c5=8

14) {sn}=3n

4n+5

A) s1=1

3,s2=3

7,s3=9

23,s4=9

29,s5=81

343 B) s1=1

3,s2=6

13,s3=9

17,s4=4

7,s5=3

5

C) s1=3

7,s2=9

23,s3=9

29,s4=81

343 ,s5=243

1367 D) s1=6

13,s2=9

17,s3=4

7,s4=3

5,s5=18

29

15) {cn}=(–1)n

(n+5)(n+1)

A) c1=–1

12,c2=1

21,c3=–1

32,c4=1

45,c5=–1

60

B) c1=1

12,c2=–1

21,c3=1

32,c4=–1

45,c5=1

60

C) c1=–1

8,c2=1

10,c3=–1

12,c4=1

14,c5=–1

16

D) c1=1

8,c2=–1

10,c3=1

12,c4=–1

14,c5=1

16

16) {cn}=4n

n

A) c1=4,c2=8,c3=64

3,c4=64,c5=1024

5B) c1=0,c2=4,c3=8,c4=64

3,c5=64

C) c1=4

,

c2=16

,

c3=64

,

c4=256

,

c5=1024 D) c1=1,c2=4

,

c3=16

,

c4=64

,

c5=256

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

Page2

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) 2

,

10

,

18

,

26

,

34

,

…

A) an=2(4n–3 ) B) an=8n–3C)a

n=2(8)

n–1D) an=6n–8

20) –1

,

1

,

3

,

5

,

7

,

…

A) an=2n–3B)a

n=3n–2C)a

n=–1(2)

n–1D) an=n+2

21) 0,2,6,12,20,…

A) an=n2–nB)a

n=2n –2C)a

n=4n –6D)a

n=2n–1–1

22) 4

,

16

,

64

,

256

,

1024

,

…

A) an=4nB) an=12n

C) an=4+12(n–1) D) an=4n–1+3

23) 5

,

–10

,

15

,

–20

,

…

A) an=(–1)n+1·5n B) an=(–1)n·5n C) an=(–1)n+1·5D)a

n=(–1)n·5

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

Solve.

27) Thenumberofstudentsinaschoolinyearnisestimatedbythemodelan=7n2+12n+82.Abouthow

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

A) 101

,

134

,

181 B) 113

,

146

,

193 C) 108

,

134

,

181 D) 101

,

134

,

160

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

are$240,000,thenwhataretheprofitsforeachofthefirstfouryears?

A) $15,000

,

$30,000

,

$60,000

,

$120,000 B) $15,000

,

$30,000

,

$60,000

,

$150,000

C) $16,000

,

$32,000

,

$64,000

,

$126,000 D) $15,000

,

$30,000

,

$45,000

,

$60,000

Page3

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=4;an=an–1–1

A) a1=4

,

a2=3

,

a3=2

,

a4=1B)a

1= –1

,

a2= –2

,

a3=–3

,

a4=–4

C) a1=4

,

a2=5,a3=6,a4=7D)a

1=4

,

a2=5 ,a3=4,a4=3

2) a1=4;an=4an–1

A) a1=4

,

a2=16

,

a3=64

,

a4=256 B) a1=4

,

a1=15

,

a2=14

,

a4=13

C) a1=12

,

a1=48

,

a2=192

,

a4=384 D) a1=4

,

a1=18

,

a2=66

,

a4=258

3) a1=3;an=2an–1–3

A) a1=3

,

a2=3

,

a3=3

,

a4=3B)a

1=3

,

a2=3

,

a3=9

,

a4=21

C) a1=3

,

a2=6

,

a3=12

,

a4=24 D) a1=3

,

a2=9

,

a3=21

,

a4=45

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=14;an=n–an–1

A) a1=14

,

a2=–12

,

a3=15

,

a4=–11 B) a1=14

,

a2=16

,

a3=–13

,

a4=17

C) a1=14

,

a2=28

,

a3=–14

,

a4=42 D) a1=13

,

a2=16

,

a3=14

,

a4=17

7) a1=2;an=an–1

n+1

A) a1=2,a2=2

3,a3=1

6,a4=1

30 B) a1=2,a2=1,a3=1

3,a4=1

12

C) a1=2,a2=2,a3=2

3,a4=1

6D) a1=2,a2=2,a3=1,a4=1

3

8) a1=w;an=an–1+U

A) a1=w

,

a2=w+U

,

a3=w+2U

,

a4=w+3U B) a1=w

,

a2=U

,

a3=2U

,

a4=3U

C) a1=w

,

a2=w–U

,

a3=w–2U

,

a4=w–3U D) a1=U

,

a2=U+w

,

a3=U+2w

,

a4=U+3w

9) a1=10;an=10an–1

A) a1=10,a2=10 10,a3=10 10 10,a4=10 10 10 10

B) a1=10,a2=10,a3=10 10,a4=100

C) a1=10,a2=10 10,a3=100 10,a4=1000 10

D) a1=10,a2=10,a3=10,a4=10

Page4

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.

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.

Page5

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+3)

∑

A) 4+5+6+…+(n+3) B) n+3

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

2)

n

k=1

(2k+7)

∑

A) 9+11+13+…+(2n+7) B) 2n+7

C) 7+9+11+…+(2n+7) D) 1+2+3+…+n

3)

n

k=1

(k+1)2

∑

A) 4+9+16+…+(n+1)2B) (n+1)2

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

4)

n–1

k=0

(4k+8)

∑

A) 8+12+16+…+(4n+4) B) 12 +16 +20 +…+(4n+4)

C) 8+12+16+…+(4n+8) D) 12 +16 +20+…+(4n+8)

5)

n

k=0

4

7

k

∑

A) 1+4

7

+16

49

+…+4

7

nB) 4

7

+16

49

+64

343

+…+4

7

n

C) 0+4

7

+16

49

+…+4

7

nD) 0+1+2+3+…+n

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

Page6

7)

n

k=1

k2

2

∑

A) 1

2

+2+9

2

+…+n2

2B) n2

2

C) 2+8+18+…+n2

2D) 1+2+3+…+n

8)

n

k=1

4k+1

∑

A) 42+43+44+…+4n+1B) 4+42+43+…+4n

C) 42+43+44+…+4nD) 4+42+43+…+4n+1

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

9) 2+4+6+…+16

A)

8

k=1

2k

∑

B)

8

k=0

2k

∑

C)

8

k=1

k2

∑

D)

8

k=1

2k2

∑

10) 25+35+45+…+75

A)

7

k=2

k5

∑

B)

7

k=1

k5

∑

C)

n

k=2

k5

∑

D)

7

k=3

(k–1)5

∑

11) 2+8+18+...+50

A)

5

k=1

2k2

∑B)

5

k=0

2k2

∑C)

5

k=1

k2

∑D)

5

k=1

22k

∑

12) 1

4

+2

5

+1

2

+…+13

16

A)

13

k=1

k

k+3

∑

B)

13

k=0

k

k+3

∑

C)

n

k=1

k

k+3

∑

D)

13

k=3

k

k+1

∑

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) 2

e

+4

e2

+6

e3…+2n

en

A)

n

k=1

2k

ek

∑B)

n

k=0

2k

ek

∑C)

n

k=1

2+k

ek

∑D)

n

k=0

2+k

ek

∑

Page7

15) 3

4

–9

16

+27

64

–…+(–1)(11+1)3

4

11

A)

11

k=1

(–1)(k+1)

∑3

4

k

B)

12

k=1

(–1)(k+1)

∑3

4

k

C)

n

k=1

(–1)(k+1)

∑3

4

k

D)

11

k=1

3

4

k

∑

16) 311

85

+ 310

86

+39

87

+…+34

812

A)

8

k=1

312–k

84+k

∑B)

4

k=1

312 +k

84+k

∑C)

8

k=1

312 –k

8k

∑D)

8

k=1

3–k

84+k

∑

17) 32+63+94+…+249

A)

8

k=1

(3k)k+1

∑B)

8

k=1

(3k)k

∑

C)

8

k=1

2(k–1)k+1

∑D)

8

k=1

3k2k–1

∑

18) 1

x

+u

2x

+u2

3x

+…+un–1

nx

A)

n

k=1

uk–1

kx

∑B)

n

k=1

uk

kx

∑C)

n

k=0

uk–1

kx

∑D)

n

k=0

<b>k

kx

∑

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)

12

k=1

2

∑

A) 24 B) 2 C) 12 D) 10

2)

8

k=1

k

∑

A) 36 B) 8 C) 7 D) 56

Page8

3)

7

k=4

2k

∑

A) 44 B) 14 C) 22 D) 30

4)

5

k=1

(k–9)

∑

A) –30 B) –4C)

–12 D) –26

5)

6

k=3

(4k–5)

∑

A) 52 B) 45 C) 57 D) 31

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–1)

∑

A) 47 B) 50 C) 21 D) 9

9)

4

k=2

k(k–12)

∑

A) –79 B) –90 C) –52 D) –18

10)

4

k=1

–1

3

k

∑

A) –20

81 B) 20

81 C) –16

81 D) 40

81

Page9

11)

4

k=1

(–1)k·–9k

∑

A) –18 B) 90 C) –90 D) 6561

13.2 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) 3

,

7

,

11

,

15

,

19

,

…

A) Arithmetic B) Notarithmetic

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

7) {7–5n}

A) d=–5;2

,

–3

,

–8

,

–13 B) d= –5;–5

,

–10

,

–15

,

–20

C) d=5;2

,

7,12,17 D) d= –5;2

,

–1 ,–6,–11

8) {6n+5}

A) d=6;11

,

17

,

23

,

29 B) d=6;6

,

11

,

17

,

23

C) d=5;11

,

17

,

23

,

29 D) d=5;6

,

11

,

17

,

23

9) 1

4

+n

8

A) d=1

8;3

8,1

2,5

8,3

4B) d=1

4;1

4,3

8,1

2,5

8C) d=1

8;1

4,3

8,1

2,5

8D) d=1

4;3

8,1

2,5

8,3

4

Page10

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=6;d=9

an=?;a16=?

A) an=–3+9n;a16=141 B) an=6+9n;a16 =141

C) an=–3+9n;a16=69 D) an= –3–9n;a16=141

2) a=92;d=–6

an=?;a10=?

A) an=98–6n;a10=38 B) an=92 –6n;a10=38

C) an=98–6n;a10=14 D) an=92 –6n;a10=14

3) a=9;d=–2

an=?;a13=?

A) an=11–2n;a13=–15 B) an=9–2n;a13 =–15

C) an=11–2n;a13=–9D)a

n=11 +2n;a13=–15

4) a=–9;d=4

an=?;a6=?

A) an=–13+4n;a6=11 B) an= –9+4n;a6=11

C) an=–13+4n;a6=43 D) an=–13 –4n;a6=11

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

5) Thetwenty–fourthtermofthearithmeticsequence–4

,

–1

,

2

,

…

A) 65 B) 68 C) –73 D) –76

6) Theeighteent

h

termofthearithmeticsequence0,12

,

24

,

…

A) 204 B) 216 C) 228 D) 187

7) Theelevent

h

termofthearithmeticsequence29

,

25

,

21

,

…

A) –11 B) –15 C) –40 D) 69

8) Thethirteenthtermofthearithmeticsequence14 3,93,43,…

A) –46 3 B) –51 3 C) 74 3 D) 79 3

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

9) 9thtermis29;14thtermis–6

A) a1=85

,

d=–7

,

an=an–1–7B)a

1=85

,

d=7

,

an=an–1+7

C) a1=92

,

d=–7

,

an=an–1–7D)a

1=92

,

d=7

,

an=an–1+7

10) 7thtermis–48;13thtermis–102

A) a1=6

,

d=–9

,

an=an–1–9B)a

1=6

,

d=9

,

an=an–1+9

C) a1=15

,

d=–9

,

an=an–1–9D)a

1=15

,

d=9

,

an=an–1+9

Page11

11) 7thtermis16;15thtermis48

A) a1=–8

,

d=4

,

an=an–1+4B)a

1= –8

,

d=4

,

an=an–1–4

C) a1=–12

,

d=4

,

an=an–1+4D)a

1= –12

,

d=4

,

an=an–1–4

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=2

,

d=–4;a8

A) –26 B) –30 C) 30 D) 34

14) a=9

,

d=4;a19

A) 81 B) 85 C) –63 D) –67

15) a=6

,

d=4;a9

A) 38 B) 42 C) –26 D) –30

16) 12

,

16

,

20

,

…;a29

A) 124 B) 128 C) –100 D) –104

17) a 17 =–108

,

a15=–96;a1

A) –12 B) –18 C) –6D)

–24

18) a 12 =10

,

a18=22;a4

A) –6B)

–4C)2 D)

–12

19) a 35 =45

2,a21=79

6;a4

A) 11

6B) 5

2C) –1D)

2

3

Solvetheproblem.

20) Thepopulationofatownisincreasingby400 inhabitantseachyear.Ifitspopulationatthebeginningof

1990was27,161,whatwasitspopulationatthebeginningof1996?

A) 29,561inhabitants B) 162,906 inhabitants

C) 29,161inhabitants D) 325,812 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

2) 1+2+3+…+207

A) 21528 B) 21321 C) 43,056 D) 42,849

3) 2+4+6+…+1730

A) 749,090 B) 748,225 C) 747,360 D) 749,956

Page12

4) 1+3+5+…+559

A) 78,400 B) 78,680 C) 78,961 D) 77,841

5) 6+12+18+…+756

A) 48006 B) 47628 C) 48387 D) 47250

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)

25

n=1

(6n+5)

∑

A) 2075 B) 2000 C) 2175 D) 2262.5

8)

50

n=1

(4n–7)

∑

A) 4750 B) 4650 C) 4875 D) 5050

9)

28

n=1

(–3n+6)

∑

A) –1050 B) –1008 C) –966 D) –868

10)

28

n=1

(–3n–4)

∑

A) –1330 B) –1288 C) –1260 D) –1162

11)

10

n=1

(3.4n+8.75)

∑

A) 274.5 B) 42.75 C) 231.75 D) 192.4

Solve.

12) Atheaterhas26rowswith29seatsinthefirstrow,32 inthesecondrow,35 inthethirdrow,andsoforth.

Howmanyseatsareinthetheater?

A) 1729seats B) 1768 seats C) 3458 seats D) 3536 seats

13) Abrickstaircasehasatotalof18stepsThebottomsteprequires139 bricks.Eachsuccessivesteprequires5

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

A) 1737bricks B) 3267 bricks C) 3474 bricks D) 1692 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

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

Page13

13.3 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}=21

2

n–1

A) an=21

2

n–1

r=1

2;s1=2,s2=1,s3=1

2,s4=1

4

B) an=2·(2)n–1

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

C) an=21

4

n–1

r=1

4;s1=2,s2=1

2,s3=1

8,s4=1

32

D) an=1

2

(2)n–1

r=1

2;s1=2,s2=4,s3=8,s4=16

2) {sn}={4n}

A) r=4;s1=4

,

s2=16

,

s3=64

,

s4=256 B) r=4;s1=4

,

s2=8

,

s3=12

,

s4=16

C) r=4n;s1=4

,

s2=16

,

s3=64

,

s4=256 D) r=4n;s1=4

,

s2=8

,

s3=12

,

s4=16

3) {tn}=1

4

n

A) r=1

4;t1=1

4,t2=1

16,t3=1

64,t4= 1

256 B) r=1

4n;t1=1

4,t2=1

16,t3=1

64,t4=1

256

C) r=1

4;t1=1

4,t2=1

4,t3=1

4,t4=1

4D) r=1

4n;t1=1

4,t2=1

4,t3=1

4,t4=1

4

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}={42n}

A) r=16;s1=16

,

s2=256

,

s3=4096

,

s4=65,536 B) r=4;s1=16

,

s2=256

,

s3=4096

,

s4=65,536

C) r=8;s1=8

,

s2=16

,

s3=24

,

s4=32 D) r=16;s1=8

,

s2=16

,

s3=24

,

s4=32

6) {un}=2n

5n–1

A) r=2

5;u1=2,u2=4

5,u3=8

25,u4=16

125 B) r=2

5;u1=2

5,u2=4

25,u3=8

125 ,u4=16

625

C) r=2;u1=2,u2=4

5,u3=8

25,u4=16

125 D) r=2;u1=2,u2=4

5,u3=8

5,u4=16

5

Page14

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) {5n–4}

A) Arithmetic;d=5 B) Geometric;r=5 C) Arithmetic;d= –4 D) Neither

8) {3n2}

A) Geometric;r=3

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

9) 3

2

n

A) Geometric;r=3

2B) Arithmetic;d=3

2C) Geometric;r=2

3D) Neither

10) {4n2–3}

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

11) 9–1n

A) Arithmetic;d=–1 B) Geometric;r= – 1

C) Arithmetic;d=9 D) Neither

12) {7n

/

7}

A) Geometric;r=71

/

7B) Arithmetic;d=71

/

7

C) Geometric;r=7 D) Neither

13) 6

,

8

,

11

,

15

,

…

A) Geometric;r=6 B) Arithmetic;d=6 C) Geometric;r=36 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

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=6;r=3

A) a5=486;an=6·(3)n–1B) a5=486;an=6·(3)n

C) a5=1458;an=6·(3)nD) a5=1458;an=6·(3)n–1

2) a=5;r=1

2

A) a5=5

16;an=5·1

2

n–1B) a5=5

16;an=5·1

2

n

C) a5=5

32;an=5·1

2

n–1D) a5=5

32;an=5·1

2

n

Page15

3) a=4;r=–4

A) a5=1024;an=4·(–4)n–1B) a5=1024;an=4·(–4)n

C) a5=–256;an=4·(–4)n–1D) a5=–256;an=4·(–4)n

4) a=–3;r=–5

A) a5=–1875;an=–3·(–5)n–1B) a5=–1875;an=–3·(–5)n

C) a5=375;an=–3·(–5)n–1D) a5=375;an=–3·(–5)n

5) a=4;r=2π

A) a5=64π4,an=4·2n–1πn–1B) a5=128π5,an=4·2nπn

C) a5=64π,an=4·2n–1πD) a5=4+8π

,

an=4+2π(n–1)

6) a=7;r=7

A) a5=49 7,an=7n/2 B) a5=16,807,an=7n

C) a5=2401 7,an=7n–1/2 D) a5=77,an=7n/2–1

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

7) 6thtermof1,1

2,1

4,…

A) a6=1

32 B) a6=1

2C) a6=1

16 D) a6=1

128

8) 5thtermof1,2

,

4

,

…

A) a5=16 B) a5=64 C) a5=32 D) a5=4

9) 8thtermof–1,3

,

–9

,

…

A) a8=2187 B) a8= –2187 C) a8= –6561 D) a8=6561

10) 7thtermof0.7

,

0.07

,

0.007

,

...

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

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

11) a=–4;r=–5

A) an=–4·(–5)n–1B) an=–4·(–5)nC) an=–4·5nD) an=–5·(–4)n–1

12) a=–3;r=–1

2

A) an=–3–1

2

n–1B) an=–3–1

2

nC) an=–31

2

n–1D) an=–1

2

(–3)n–1

13) 6,3,3

2,3

4,…

A) an=61

2

n–1B) an=6·2n–1C) an=61

4

n–1D) an=1

2

(6)n–1

14) 8

,

16

,

32

,

64

,

128

,

…

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

Page16

15) –2

,

–4

,

–8

,

–16

,

–32

,

…

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

16) 5

,

–15

,

45

,

–135

,

405

,

…

A) an=5·(–3)n–1B) an=5·(–3)nC) an=a1–3nD) an=5·(–3)n

17) 2,1

2,1

8,1

32,1

128 ,…

A) an=2·1

4

n–1B) an=2·1

4

nC) an=2·1

4

n+1D) an=2·1

16

n–1

18) 3,–3

4,3

16,–3

64,3

256 ,…

A) an=3·–1

4

n–1B) an=3·–1

4

nC) an=3·–1

4

n+1D) an=3·–1

16

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

22) Anewpieceofequipmentcostacompany$66,000.Eachyear,fortaxpurposes,thecompanydepreciates

thevalueby25%.Whatvalueshouldthecompanygivetheequipmentafter8years?

A) $6607 B) $1 C) $8810 D) $4

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16grams?

A) 1

8

gram B) 8grams C) 1

4

gram D) 4grams

24) Abicyclewheelrotates300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) 59times B) 89times C) 0times D) 11times

Page17

25) Apendulumbobswingsthroughanarc60 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10swings?Roundyour

answertotwodecimalplaces,ifnecessary.

A) 2.75inches B) 1.95 inches C) 1.39 inches D) 383.4 inches

26) Abaseballplayersignsacontractwithastartingsalaryof$850,000 peryearandanannualincreaseof5%

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

A) $1,084,839 B) $1,085,545 C) $1,083,481 D) $1,087,306

27) Carlatakesajobwithastartingsalaryof$27,000 forthefirstyearwithanannualincreaseof5%beginning

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

A) $36,183 B) $36,797 C) $34,808 D) $38,621

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

2

3(4)k

∑

A) 2728

3B) 2707

3C) 2734

3D) 2779

3

3)

3

k=1

4

5

k+1

∑

A) 976

625 B) 244

125 C) 196

125 D) 5904

625

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) 6+18+54+162+486+…+6·310

A) 531,438 B) 531,418 C) 531,475 D) 531,440

6) –4–12–36–108–324–…–4·313

A) –9,565,936 B) –9,565,956 C) –9,565,899 D) –9,565,934

7) 6–18+54–162+486–…+6·(–3)8

A) 29,526 B) 29,524 C) 29,533 D) 29,520

Page18

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

4(4)k

∑

A) 5456 B) 10,280 C) 272 D) 1360

10)

5

k=1

3(–3)k

∑

A) –549 B) –1845 C) 72 D) –63

11)

10

k=1

1

5

·(–2)k–1

∑

A) –68.2 B) –68.6 C) –66.8 D) –69.4

Solve.

12) Asmallbusinessownermade$40,000 thefirstyearheownedhisstoreandmadeanadditional7% 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) $49,001.72;$177,597.72 B) $13.72;$43,009.72

C) $196,520.00;$420,120.00 D) $334,084.00;$754,204.00

13) AsSuneeimprovesheralgebraskills,shetakes0.8 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) 29min;235min B) 23min;235 min C) 29 min;207 min D) 23min;207 min

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

Page19

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) 2+2

3

+2

9

+2

27

+...

A) Converges;3 B) Converges;8

3C) Converges;2

3D) Diverges

2) 1–1

4

+1

16

–1

64

+...

A) Converges;4

5B) Converges;3

4C) Converges;–1

4D) Diverges

3) 75+15+3+3

5

+...

A) Converges;375

4B) Converges;–75

4C) Converges;93 D) Diverges

4) 2+1+1

2

+…

A) Converges;4 B) Converges;2 C) Converges;1 D) Diverges

5) 1–1

3

+1

9

–…

A) Converges;3

4B) Converges;1 C) Converges;–1

3D) Diverges

6) 24+12+6+…

A) Converges;48 B) Converges;–24 C) Converges;42 D) Diverges

7) –12–3–3

4

–…

A) Converges;–16 B) Converges;4 C) Converges;–63

4D) Diverges

8) 11+15+17+19+…

A) Converges;30 B) Converges;41 C) Converges;∞D) Diverges

9)

∞

k=1

4

9

·5k–1

∑

A) Converges;20

9B) Converges;200

9C) Converges;∞D) Diverges

10)

∞

k=1

42

3

k–1

∑

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

Page20

Solve.

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

only60%asfarasonthepreviousswing.Howfarwillitswingaltogetherbeforecomingtoacomplete

stop?

A) 200inches B) 133inches C) 100 inches D) 267 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72 vibrationsinthefirstsecondandineachsecond

thereaftermakes6

7

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

makesbeforeitstopsvibrating.

A) 504vibrations B) 84vibrations C) 514 vibrations D) 79vibrations

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

Page21

3) Tosaveforretirement,youdecidetodeposit$1500 intoanIRAattheendofeachyearforthenext40

years.Iftheinterestrateis7%peryearcompoundedannually,findthevalueoftheIRAafter40years.

(Roundtothenearestdollar.)

A) $299,453 B) $20,962 C) $278,460 D) $4,618,809

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

5.5%peryearcompoundedannually.Youplantoinvest$2500attheendofeachyearforthenext25

years.Howmuchwillyouraccounthaveinitattheendof25years?

A) $127,881 B) $129,651 C) $126,338 D) $129,179

5) Lonniedeposits$125eachmonthintoanaccountpayingannualinterestof6.5%compoundedmonthly.

Howmuchwillhisaccounthaveinitattheendof5years?

A) $8834 B) $712 C) $8680 D) $8963

6) Laurainvests$300eachquarterinafixed–interestmutualfundpayingannualinterestof5.5%

compoundedquarterly.Howmuchwillheraccounthaveinitattheendof15years?

A) $27,689 B) $6723 C) $83,470 D) $27,818

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.

13.4 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) 2+7+12+…+(5n–3)=n

2(5n–1)

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)

Page22

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·9+2·9+3·9+...+9n=9n(n+1)

2

15) 48n=48n

13.5 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

6

A) 924 B) 665,280 C) 462 D) 2

2) 7

2

A) 21 B) 7 C) 0 D) 1

3) 12

6

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

Page23

4)

143

3

A) 477,191 B) 2,863,146 C) 143!

140! D) 140

5)

8

0

A) 1 B) 40,320 C) 2 D) 0

6) 4

1

A) 4 B) 1 C) 4

3D) 4!

7)

11

A) 1 B) 39,916,800 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–7)5

A) x5–35x4+490x3–3430x2+12,005x–16,807 B) x5–35x4+490x3–3430x2+12,005x–7

C) x5–35x4+980x3–6860x2+12,005x–16,807 D) x5–35x4+980x3–6860x2+12,005x–7

3) (3x+5)3

A) 27x3+135x2+225x+125 B) 27x3+135x2+135x+125

C) 9x6+15x3+15,625 D) 9x2+30x+25

4) (4x–1)4

A) 256x4–256x3+96x2–16x+1B)

–256x4+256x3+96x2+16x+1

C) (16x2–4x+1)4D) 256x3–256x2+96x–16

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) (2x2+3)3

A) 8x6+36x4+54x2+27 B) (4x4+12x2+9)3

C) 8x3+36x2+54x+27 D) 16x8+8x6+36x4+54x2+27

7) (4x–5y)3

A) 64x3–240x2y+300xy2–125y3B) 64x3–80x2y+100xy2–125y3

C) 16x3y–20x2y2+25xy3D) 16x3y–40x2y2+25xy3

8) (x2+2y)4

A) x8+8x6y+24x4y2+32x2y3+16y4B) x8+2x6y+24x4y2+16x2y3+16y4

C) x4+8x3y+24x2y2+32xy3+16y4D) x8+8x6y+24x4y2+8x2y3+16y4

9) (x5+4y)4

A) x20+16x15y+96x10y2+256x5y3+256y4B) x9+12x8y+96x7y2+192x5y3+256y4

C) x9+16x8y+24x7y2+16x5y3+4y4D) x20+12x15y+96x10y2+192x5y3+256y4

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

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+5)4

A) x2+45

x3/2+30x+20 5x1/2+25 B) x2+25x3/2+15x+10 5x1/2+25

C) x2+45

x1/2+30x+20 5x1/2+625 D) x2+25

15) (fx+gy)5

A) f5x5+5f4gx4y+10f3g2x3y2+10f2g3x2y3+5fg4xy4+g5y5

B) f5x5+5f3g2x3y2+5fg4xy4+g5y5

C) f5x5+5fgx4y+10f3g2x3y2+10f2g3x2y3+5fgxy4+g5y5

D) f5x5+g5y5

Page25

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

16) Thecoefficientofxintheexpansionof(3x+7)3

A) 441 B) 189 C) 49 D) 882

17) Thecoefficientofxintheexpansionof(2x+4)5

A) 2560 B) 1280 C) 5120 D) 640

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(3x+4)3

A) 144x B) 108x2C) 16 D) 288x

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

A) 320x B) 640x2C) 160 D) 160x

23) The6thtermintheexpansionof(4x+3y)9

A) 7,838,208x4y5B) 2,612,736x4y6C) 2,612,736x5y4D) 1,959,552x5y4

24) The8thtermintheexpansionof(x+3y)12

A) 1,732,104x5y7B) 577,368x5y8C) 577,368x7y5D) 1,732,104x7y5

25) The11thtermintheexpansionof(4x+2y)12

A) 1,081,344x2y10 B) 540,672x2y11 C) 540,672x10y2D) 270,336x10y2

26) The4thtermintheexpansionof(x–3y)12

A) –5940x9y3B) 1980x9y4C) 1980x3y9D) –5940x3y9

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

Ch.13 Sequences;Induction;theBinomialTheorem

AnswerKey

13.1 Sequences

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

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

3 UseSummationNotation

4 FindtheSumofaSequence

13.2 ArithmeticSequences

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

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

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

13.3 GeometricSequences;GeometricSeries

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

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

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

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

5 SolveAnnuityProblems

13.4 MathematicalInduction

1 ProveStatementsUsingMathematicalInduction

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13.5 TheBinomialTheorem

1 EvaluateaBinomialCoefficient

2 UsetheBinomialTheorem