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• Timeallowed:45minutes• CAScalculatorsmaybeused.• PartA:5multiple-choicequestions(5marks)• PartB:9short-answerquestions(28marks)• PartC:4analysisquestions(18marks)

Part AMultiple-choice questions

5multiple-choicequestions1markeach:5marks

Circlethecorrectanswer.1 Atthebeginningofaparticulardaythetemperaturewas10°C.Itbecamesteadilywarmerduringthe

day,increasing0.6°Ceveryquarterofanhour.Thetemperaturesoveraperiodof5hoursareshownbelow.

5 10 15 20t (1/4 hour)

T (8C)

10

20

30

40

Afterhowmanyhours,tothenearesthour,isthetemperature20°C?

A 2

B 4

C 5

D 17

E 16

[1 mark]

Chapter 3: Arithmetic and geometric sequences

TopiC TEsT 2

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2 Thefirstfourtermsofasequenceare2,8,14,20,26.The10thtermofthesequenceis:

A 2

B 32

C 56

D 50

E 290

[1 mark]

3 Thefirstfivetermsofasequenceare2,8,14,20,26.Thesumtothe10thtermofthesequenceis:

A 320

B 32

C 56

D 50

E 290

[1 mark]

4 Thecommondifferenced5tn2tn21forthesequence1,28,217,226,235,…is:

A 9

B 29

C 10

D 210

E 244

[1 mark]

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5 Thecommonratio rt

tn

n 15

2

forthesequence1,24,16,264,…is:

A 9

B 29

C 4

D 24

E 256

[1 mark]

Part B Short-answer questions

9short-answerquestions28marks

Showyourworkingwhereappropriate.1 Findthearithmeticsequenceifthesecondterminthesequenceis3andtheninthtermis210.

[4 marks]

2 Ifthefirstsixtermsofasequenceare2,2.5,3,3.5,4,4.5,statethesumofthesequenceto10terms.

[3 marks]

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3 Findthesumofthearithmeticsequence30,27,24,21,…,2111.

[4 marks]

4 Calculatethesumofallthemultiplesof6between1and100.

[4 marks]

5 Ifx,x13,andx15arethefirstthreetermsofageometricsequence,calculatethevalueofx.

[2 marks]

6 Findthefirstfourtermsofthesequencedescribedbytherecursion:

a156andan523an21.

[2 marks]

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7 Ifthefirstfivetermsofasequenceare22,6,218,2243,729,22187,statethevalueofthe17thtermofthesequence.

[3 marks]

8 Findthecommonratior5t

tn

n21

forthesequence1, 13

, 19

, 127

, 181

2 2 ,…

[2 marks]

9 Findthegeometricsequenceifthesecondterminthatsequenceis3andtheninthtermis354.

[4 marks]

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Part CAnalysis questions

4analysisquestions18marks

Showyourworkingwhereappropriate.1 Jamestakesoutapersonalloanof$8500tobuyasecond-handcar,andischarged9%everyyearin

interestontheremainingbalance.Jamesisunabletopayanythingoffthebalanceofhisloan.

a TabulatethebalanceofJames’loanoveraperiodof11years.

b AfterhowmanyyearsdoesJamesowe$20 000?

[4 marks]

2 Findavalue(s)ofxsuchthat 1 ,4

,6

, ...2

3

xx x willformsuccessivetermsofageometricsequence.

[4 marks]

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3 Curtainsareinstalledinlayersinahome.Onelayerletsin95%oflight.

a Whatpercentageoflightisletinby:

i twolayers

ii threelayers

iii 10layersofcurtaining?

b Howmanylayerswillletin30%oflight?

[5 marks]

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4 Aballisdroppedfromaheightof1metreandbouncesupto 23ofitsoriginalheight.Itcontinues

bouncing,rising 23ofitsheightoneachbounceuntiliteventuallyreachestheground.

a Whatisthetotaldistancethroughwhichittravels?

b Willtheballeverstopbouncing?

[5 marks]

[ Total marks: 51]

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Answers part A – Multiple-choice questions [1 mark each]

1 a510,d50.6,sotn5101(n21)(0.6)

2051020.610.6n

n510.60.6

517.7quarterhours

17.74

54.4hrs(roundto4)

[B

2 a52

d582256

tn5216(n21)

t105216(1021)

521639

556[C

3 a52

d582256

Usetheformula:

Sn5n2

[2a1(n21)d]

S105102

[41936]

55(4154)

5290

[E

4 d5tn2tn21

52821

52172(28)

52262(217)

52352(226)

529

[B

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5 r5t

tn

n 12

52 4

15

1642

524

[D

part B – short-answer questions

1 t253andt95210 [1 mark]

Usetn5a1(n21)dtosolvesimultaneouslyforaandd.

a1(221)d53anda1(921)d5210 [1 mark]

givesa5 347

,d52 137

[1 mark]

Thearithmeticsequenceis 347

, 217

, 87

, ... [1 mark]

2 Forthesequence2,2.5,3,3.5,4,4.5,a52,d50.5. [2 marks]

Use Sn5 n2

[2a1(n21)d]tofindS10.

S105102

[2(2)1(1021)0.5] [1 mark]

[S10542.5

3 Forthearithmeticsequence30,27,24,21,…,2111,a530,d523,l52111. [2 marks]

UseSn5n2

(a1l)wherel5tn5a1(n21)d.

Solveforn,l5tn521115301(n21)(23) gives n 548. [1 mark]

[S485 482

(302111) [1 mark]

[S48521944

4 Arithmeticsequenceis6,12,18,24,…,96,wherea56,d56,l596. [2 marks]

Weneedtochoose96,becausethatisthelastmultipleof6before100.

UseSn5n2

(a1l)wherel5tn5a1(n21)d.

Solveforn,l5tn596561(n21)(6) gives n 516. [1 mark]

[S165 162

(6196) [1 mark]

[S165816

Theothersumformulacouldalsobeused,usingn516.

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5 Usingr5 tt

tt

tt

2

1

3

2

4

3

5 5 5 ... givesr5 xx

xx

1

15

153

3 . [1 mark]

x(x15)5(x13)(x13)

x215x5x216x19

052x19

x529 [1 mark]

6 Sequenceis6,218,54,2162 [2 marks]

7 Sequenceis22,6,218,2243,729,22187,wherea522,r523. [2 marks]

Usingtn5arn21tofindt17

t175(22)(23)1721 [1 mark]

[t175286 093 442

8 Usingr5t

tn

n21tofindr5

213

1 [1 mark]

Givesr5 213. [1 mark]

9 t253andt95354 [1 mark]

tn5arn21tosolvesimultaneouslyforaandr.

ar 22153andar9215354 [1 mark]

givesa5 31187

,r5 1187 [1 mark]

Thegeometricsequenceis 3118

, 3, 3 118, ...7

7 [1 mark]

part C – Analysis questions

1 a Year Loan balance1 85002 96253 100994 110085 119996 130797 142568 155399 1693610 1846011 20121

[2 marks]

b Jamesowes$20000after11years. [2 marks]

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2 Using 5 5 5 5r tt

tt

tt

...2

1

3

2

4

3

gives 5 5r

x

x

x

x

6

4

41

3

2

[2 marks]

r5 x3

63 4

x5 x

43 x 2

1

46

3xx

5 x3

4

434x35x336x

16x356x4

853x

x583 [2 marks]

3 a i Ifonelayerofcurtainletsin95%oflightwecansaythat:

0.95n5x,wheren51curtainlayerandx5percentageoflightthatisletin.

Ifwehavetwocurtainlayers,thenn52,soweget:

0.9525x

x50.9025

[Twocurtainlayersletin90.25%oflight. [1 mark]

ii0.9535x

0.95350.8574

[Threecurtainlayersletin85.74%oflight. [1 mark]

iii 0.95105x

0.951050.5987

[Tencurtainlayersletin59.87%oflight. [1 mark]

b Solve0.95n50.3

[n523

Therefore,23layerswillbeneededtoblockout30%oflight. [2 marks]

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4 a Use S ar∞ 5

21witha5

23,r5

23. [2 marks]

Distancetravelledisoriginal1metreplustwicetheheightofeachsubsequentbounce.

Totaldistancetravelled5

1 21

1 2

23

1 23

12

5 12

5

ar

1 2

2313

5

1

5

[2 marks]

55metres

b Asthisisalimitingsum,intheory,theballwillneverstopbouncing. [1 mark]

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