Sequences an Seres - Nulake 2.3 Sample.pdf · Using Spreadsheets for Sequences and Series ... 1st...

Year 12Mathematics

Contents

uLake Ltdu a e tduLake LtdInnovative Publisher of Mathematics Texts

Robert Lakeland & Carl NugentSequences and Series

• AchievementStandard .................................................. 2

• Sequences............................................................. 2

• ArithmeticSequences .................................................. 5

• SumsofanArithmeticSequence.......................................... 10

• ApplicationsofArithmeticSequences..................................... 13

• GeometricSequences..................................................... 19

• SumsofGeometricSequences............................................ 23

• SumofaGeometricSequencetoInfinity................................... 27

• ApplicationsofGeometricSequences..................................... 30

• SigmaNotation........................................................ 35

• CompoundInterest..................................................... 37

• Inflation............................................................... 41

• Depreciation........................................................... 43

• RadioactiveDecay..................................................... 45

• UsingSpreadsheetsforSequencesandSeries............................... 48

• SolvingApplicationProblemsInvolvingSequences.......................... 51

• PracticeInternalAssessment ............................................ 54

• Answers............................................................... 57

IAS 2.3

IAS 2.3 – Year 12 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent

2 IAS 2.3 – Sequences and Series

Sequences

Definitions

NCEA 2 Internal Achievement Standard 2.3 – Sequences and Series

Thisachievementstandardinvolvesapplyingsequencesandseriesinsolvingproblems.

◆ ThisachievementstandardisderivedfromLevel7ofTheNewZealandCurriculumandisrelatedto theachievementobjective ❖ usearithmeticandgeometricsequencesandseries intheMathematicsstrandoftheMathematicsandStatisticsLearningArea.◆ Applysequencesandseriesinsolvingproblemsinvolves: ❖ selectingandusingmethods ❖ demonstratingknowledgeofconceptsandterms ❖ communicatingusingappropriaterepresentations.◆ Relationalthinkinginvolvesoneormoreof: ❖ selectingandcarryingoutalogicalsequenceofsteps ❖ connectingdifferentconceptsorrepresentations ❖ demonstratingunderstandingofconcepts ❖ formingandusingamodel; andalsorelatingfindingstoacontextorcommunicatingthinkingusingappropriatemathematical statements.◆ Extendedabstractthinkinginvolvesoneormoreof: ❖ devisingastrategytoinvestigateasituation ❖ identifyingrelevantconceptsincontext ❖ developingachainoflogicalreasoning,orproof ❖ formingageneralisation andalsousingcorrectmathematicalstatements,orcommunicatingmathematicalinsight.◆ Problemsaresituationsthatprovideopportunitiestoapplyknowledgeorunderstandingof mathematicalconceptsandmethods.Situationswillbesetinreal-lifeormathematicalcontexts.◆ Methodsincludeaselectionfromthoserelatedto: ❖ thegeneraltermofasequence ❖ apartialsumofasequence ❖ thesumtoinfinityofageometricseries ❖ findingthevalueofthefirstterm,commondifferenceorcommonratioofasequence ❖ findingthenumberoftermsinasequence.◆ Methodscouldrequiresolvingequations,whichcouldinvolveusinglogarithms.

Achievement Achievement with Merit Achievement with Excellence• Applysequencesandseriesin

solvingproblems.• Applysequencesandseries,

usingrelationalthinking,insolvingproblems.

• Applysequencesandseries,usingextendedabstractthinking,insolvingproblems.

Asequenceisasuccessionofnumbersseparatedusuallybycommas.Examplescouldbe: 4,7,10,13,...or 8,16,32,64...

Eachmemberofthesequenceiscalledatermandmayberepresentedbythelettert(forterm)withasubscriptbeingitspositioninthesequence.i.e. t1=1stterm

t2=2ndtermetc.



Example

Findthenumberoftermsinthearithmeticsequence

23,29,35,41,47,...,125.

tn =a+(n–1)d

tn =23+(n–1)6substituting

tn =6n+17simplifying

Tofindthenumberoftermsinthesequencewesettheformulaforthenthtermequaltothelasttermgiveninthesequenceandsolveforn.

125 =6n+17

6n =125–17

n =18

Thereforethereare18termsinthesequence.

Usinga=23(firstterm)andd=6(commondifference)wesubstituteintothegeneralformulaforanarithmeticsequence.

Example

Thesecondtermofanarithmeticsequenceis28andthe6thterm0.Findthefirstfourtermsofthesequence.

Lookingattheinformationwehavebeengivenwecanrepresentthesequenceasfollows.

1st 2nd 3rd 4th 5th 6th ...a, 28, 28+d, 28+2d, 28+3d, 28+4d, ...Yetthe6thterm=0soT6=0gives 28+4d=0

d=–7

Constantdifference=–7sothefirst4terms

1st 2nd 3rd 4th and 5th 6th ...35, 28, 21, 14, 7, 0, ...

Example

Anarithmeticsequencehas5thtermof24and7thtermof40.Whatarethefirstthreetermsandtheruleforthesequence?

2d =40–24 d =8Therefore tn =a+(n–1)8Weknowthe5thtermis24thereforesubstitutingthisintheruleaboveweget

24 =a+(5–1)8

24 =a+32

a =–8

Therefore tn =–8+(n–1)8

=8n–16Substitutingn=1,2and3forthefirstthreetermsgivest1=

–8,t2=0andt3=8.

Betweenthe5thand7thtermswehavetwoconstantdifferencesso

ExampleFindtheformulaforthenthtermand20thtermofthearithmeticsequence

5,9,13,17,21,...

tn =a+(n–1)d

tn =5+(n–1)4 substituting

tn =4n+1 simplifying

Tofindthe20thtermwesubstituten=20

t20 =4(20)+1

t20 =81

Weknowitisarithmeticsousinga=5(firstterm)andd=4(commondifference)wesubstituteintothegeneralformula.

ExampleFindthefirstfourtermsofthearithmeticsequences.

a) a=2,d=–5 b) a=–4.5,d=1.2

a) 2,2–5,2–5–5,2–5–5–5 2,–3,–8,–13b) –4.5, –4.5+1.2,–4.5+1.2+1.2, –4.5+1.2+1.2+1.2 –4.5,–3.3,–2.1,–0.9

ExampleFindthegeneraltermforthearithmeticsequencewitha=4andd=–3.

tn=a+(n–1)dtn=4+(n–1)

–3tn=

–3n+3+4 tn=

–3n+7

©uLake LtduLake Ltd Innovative Publisher of Mathematics Texts

13IAS 2.3 – Sequences and Series


Applications of Arithmetic Sequences

Applications of Arithmetic Sequences

Inthissectionwelooktoapplyourknowledgeofarithmeticsequencestorealproblemsincommonsituations.Likeanyapplicationorwordtypeproblem,itisimportanttoidentifywhatisactuallybeingaskedandtousetheappropriateformula(e).Sometimesdrawingadiagramorjustlistingdownthesequenceofnumbersthataregiven,canhelptoclarifytheproblem.Withapplicationproblemsthatinvolvearithmeticsequencesitisimportanttoidentifythekeyparametersofthesequencesuchasthefirsttermandcommondifferencebeforeproceeding.

a) Sequenceofdrawersis5,10,15,20,...

Using Sn=n

22a+ n −1( )d

S13=13

22 5( )+ 13−1( )5

S13=455drawers

b) Thenextcolumnwouldhold14cabinetsand theoneafterthat15etc.

Since14+15+16+17=62,

thereforeitwouldaddanadditional fourcolumns.

Example

Acompanysellsastoragestackingsystemcomprisingcabinets,thatcanfitontopofoneanother.Eachcabinetcontains5slidingdrawers.

a) Acustomerhasdecidedtousethesamearrangementtothatdepictedinthediagram(i.e.column1hasonecabinetof5drawers,thencolumn2hastwocabinetseachof5drawersetc.)

Howmanydrawersintotalwouldshehavewitha13columnarrangement?

b) Ifthesamecustomerasina)purchasedanadditional65cabinets,howmanymorecompletedcolumnsdoesitaddtoherexistingarrangement?




86. Annhasbeensavingforherretirement.Shehassaved$89640after24years.

Eachyearshehasincreasedhersavingsfromthepreviousyearbythesameamount.InYear10shesaved$3210.HowmuchmoneydidAnnsaveinthefirstyearofhersavingsplan?

Excellence–Investigatesolutionstothefollowingproblems.87. Acompanymanufacturingaplasticcover

orskinfortheiPhoneknowsthatproducing500coverswillcost$9650or$19.30each.Toproduce1250coverswillcost$15837.50or$12.67each.

Marketresearchhasshownthatthecompanycanonlysellthecoversat$15each.

Therearebothfixedcostsandvariablecostswhenproducingthecovers.Thefixedcostisthesamenomatterhowmanycoversareproduced.Thevariablecostdependsdirectlyonthenumberofcoversproduced.

Usingthesefigures,whatistheminimumnumberofcoversthatmustbesoldinordertomakeaprofit?




116. Joeisnowpaid$16.00anhour.Howmuchwashepaidperhourwhenhewasfirstemployed?

117. FindtheformulathatrelatesthewageperhourthatJoeispaidtothenumberofyearshehasbeenworkingforthecompany.

118. Joeplanstoretirewhenhiswagereaches$30perhour.Joejoinedthecompanywhenhewas15yearsold.Howoldwillhebewhenheplanstoretire?

Sums of Geometric Sequences

Partial Sums of Geometric Sequences

Apartialsumisthesumofafixednumberorfinitenumberoftermsofasequence.Considerthegeometricsequence3,6,12,24,48,...Now S1 =3

S2 =3+6=9

S3 =3+6+12=21etc.TheletterSdenotesapartialsumandthesubscriptidentifiesthenumberoftermsbeingadded.ThepartialsumS8ofthesequenceaboveis

S8=3+6+12+24+48+96+192+384 ①

Alongsequencelikethiswouldbecumbersometoaddupsowelookforashortermethod.

IfwenowmultiplythepartialsumS8bythecommonratioofthesequence(i.e.2)weobtain 2S8=6+12+24+48+96+192+384+768 ➁

Sevenmembersofthisnewsequencewereintheoldequation①.Ifwenowsubtractequation①fromequation➁weobtain

2S8=6+12+24+48+96+192+384+768➁

S8=3+6+12+24+48+96+192+384 ①

S8 =–3+768 ➁–①

so S8=765(thepartialsumof8terms)

Wecannowuseasimilarapproachtoobtainageneralformulatofindthesumofntermsofageometricsequence.

ConsiderthepartialsumSnofageometricsequence.

Sn=a+ar+ar2+ar3+ar4+...+arn–1 ①

Multiplyingthepartialsumbyrthecommonratioweobtain

rSn=ar+ar2+ar3+ar4+...+arn–1+arn ➁

Subtractingequation①fromequation➁weobtain

rSn=ar+ar2+ar3+ar4+...+arn–1+arn ➁

Sn=a+ar+ar2+ar3+ar4+...+arn–1 ①

rSn–Sn =arn–a ➁–①

Sn(r–1) =a(rn–1) Factorising

Sn = ( 1)( 1)r

a rn

--

The sum of n terms of a series is often called the nth partial sum.

We obtain a series by summing the terms of a sequence. A series may be finite (i.e. a partial sum) or infinite.

The formula Sn =a(rn – 1)(r – 1)

enables us to

find the sum of n terms of a geometric sequence.

It is sometimes also written in the form

Sn =a(1 – rn )(1 – r)

Make sure you are familiar with at least one of these forms.




Applications of Geometric Sequences

Inthissectionwelooktoapplyourknowledgeofgeometricsequencesinapplicationsandcommonsituations.

Likeanyapplicationorwordtypeproblemitisimportanttoidentifywhatisactuallybeingaskedandtousethecorrectformulaorcombinationofformulae,i.e.partialsum,sumtoinfinityortheformulaforthegeneraltermtogettheanswer.Sometimesdrawingadiagramcanhelptoclarifytheproblemorjustlistingdownthesequenceofnumbersthataregivencanprovidethecatalysttosolvingit.

Applications of Geometric Sequences

ExampleApersonwhoinitiallyweighs120.0kgbeginsadiet.Thefirstmonththeylose18.0kg,thenextmonth12.0kgandthefollowingmonth8.0kgandtheycontinuetoloseweightbydecreasingbythesameratioeachsuccessivemonth.

a) Giveaformulathatrepresent,the amountofweightlostinaparticularmonth n andfindouthowmuchweighttheperson wouldbeexpectedtoloseinmonth10.

b) Howmuchwouldthepersonweighafter10monthsonthediet?

c) Ifthepersoncontinuedonthedietindefinitelywhatwouldbethemaximumamountofweighttheycouldloseandtheirfinalweight?

a) Formulafortheamountofweightlostinaparticularmonthnis

Tn =18 x 32 n 1-b l

Inmonth10theexpectedweightlostis

T10 =18 x 32 9b l

T10 =0.468kg (3sf)

b) Tofindtheamountapersonwouldweighafter10monthsonthedietweneedtofindthetotalweightlostandsubtractthisamountfromtheperson’soriginalweight.

Sn = ra r

11n

--

]^

gh

S10 = 1118

3232 10

-

-

___`

i ii j

S10 =53.1kg(1dp)

Weightofpersonafter10monthsis

120.0–53.1=66.9kg(1dp)

c) Ifthepersoncontinuedonthedietindefinitelythemaximumweightlosscanbecalculatedusingthesumtoinfinityformula.

S∞ = 118

32-_ i

S∞ =54kg

Finalweight =120–54

=66kg




ExampleTheradioactiveelementcaesiumhasahalf-lifeof30.1yearsandweinitiallystartwith100grams a) whatistherateofdecay? b) howmuchofthesubstanceisleftafter 12years? c) howlongwillittaketoreduceto5gramsof radioactivecaesium?

WeusethedecayformulawithDn=50,O=100andn=30.1,becauseweknowthatin30.1yearsonlyhalfoftheoriginalamount(100g)ofthesubstancewillexisti.e.50grams.

a) Dn =O x 1 nr100-_ i

50 =100 x 1 10030.1r-_ i

0.5 = 1 10030.1r-_ i

Wenowtakethe30.1throotofeachside.

0.9772349786 = 1− r100

( ) r =2.27650214

r =2.3%

Sothesubstanceisdecayingattherateof2.3%pa.

b) Tofindtheamountofthesubstanceleftafter12yearswesubstitutetheappropriatevaluesforO,randnintothedecayformula.

Dn =O x 1 nr100-_ i

Dn =100 x 1 – 2.3

100( )

12

Dn =76g (2sf)

After12years76gramsofthecaesiumremain.

c) Tofindthetimetoreducetheradioactivecaesiumto5gwesubstitutetheappropriatevaluesforD,O,andrintothedecayformula.

Dn =O x 1 nr100-_ i

5 =100 x 1 1002.3 n

-_ i

n =128.746

n =130years(2sf)

If we use the Solver on our graphics calculator we can put in the variables we know and solve for the unknown variable.

Using your Casio 9750GII

Using your TI-84 Plus

0 = Equationso in this case we will enter the equation in the form

0 = Dn – O x 1 100r n

-b l

Go to the Solver by pressing MATH and scrolling

down. If there is already an equation entered you will need to delete it. Hit the up arrow key and to erase the function that was previously there. Then enter the equation to solve.

It now displays your equation at the top and each variable. D – O(1 – R/100) ^ N = 0 D = 50 O = 100 R = N = 30.1

Using the cursor keys to enter in D, O and N and

place the cursor next to R and select ALPHA ENTER

to solve.

CLEAR

( – ALPHA R ÷ 11

CLEAR ALPHA D – ALPHA O

Up

0 0 ALPHA N ENTER)

To solve any equation on the TI it should be in the form

Enter the equation

Again it will list the variables and you enter all the known variables and place the cursor next to the

unknown (in this case R) and select solve F6 .

(

– ALPHA R ÷ 1 0

0 ALPHA N EXE)

Go to the Solver with MENU followed by

8

F3

to select the Solver.

SHIFTALPHA D ALPHA O=

^

1




StartingincellA2andgoingdownenterthetermnumber1,2,3,and4.NowwithyourmouseclickincellA2andsweeptocellA5.Releasethemousebuttonandmovethemousetothebottomrightcorner.Themouseiconwillchangetocrosshairs.

Selectthiscorneranddragitdownthespreadsheetuntilyouhavesufficientrows(n=15).Nowenter23incellB2andincellB3enter‘=B2+8’.InsteadoftypingB2itispossibletoclickonthecellB2toenterit.

NowclickonceonthiscellB3andreleasethemousebuttonandagainselectthebottomright

Using Spreadsheets for Sequences and Series

Wecanuseaspreadsheetforsequencestoquicklycalculatevaluesandtoadduptermsofaseries.Abigadvantageofspreadsheetsisyoucanhaveaprintedrecordofalltheresultswhichisusefulifyouareinvestigatingdifferentsequencesandneedtocompareterms.TheinstructionsgivenhereareforMicrosoftExcelbutapplysimilarlytoOpenOfficeand/orMacNumbers.Aspreadsheetisanarrayofcellswhichcanhaveaformulaeenteredintothem.Forexample,ifwehadanarithmeticsequencewewouldhead

n Term Sum

1 23 232 31 543 39 934 47 1405 55 1956 63 258

Using a Spreadsheet

Example Useaspreadsheettofindthe15thtermandthesumofthefirst15termsofthesequence23,31,39,47,55,...

Openablankspreadsheetandputthelabelsn,TermandSumaslabelsinthefirstthreecolumns.

thethreecoumnswiththeirlabelsandputthetermnumberincolumn1.Thenwecouldeithersetthefirsttermto23andtheformulainthecelldirectlyunderneathtobe

Cell=Cellabove+8

Thiscellcouldthenbecopieddownthecolumn.Forthesumswecanthenaddupthetotalofallthetermsintheseriesuptothecellweareat.ThisisexplainedinthefirstExample.Alternativelywecanentertheformulaforthegeneraltermofanarithmeticsequence.

Cell=23+(Celltotheleft–1) x 8.

corneranddragthisonecelldowntotherowwithn=15init.

InthecellC2enter=sum(B$2:B2).

TheB$2referstothestartingcellandthe$instructsthespreadsheetnottochangethis.

Afterenteringthisformulaclickonthecellandselectthebottomrightcorneranddragthisonecelldowntotherowwithn=15init.

Youshouldnowhavethefirst15termsofthesequenceandthefirst15partialsums.

Term15is135andthesumofthefirst15termsis1185.

Solutioncont...




Page 23 cont...

117. tn=8.86 x (1.03)n–1

118. 30≤8.86 x (1.03)n–1

n≥42.3

n=43

Startedworkat15sohewillretireat58years.

Page 25

119. S10=5115

120. S16=–109225

121. S8=1 128127 (1.9921875)

122. S20=67.50(4sf)

123. S16=5247(4sf)

124. S18=13.33(2dp)

125. S10=4.50(2dp)

126. S13=8.27(2dp)

127. S5001=1

128. n=13

S13= ( . )2730 31 2730 3o

129. n=7

S7=64.00(2dp)

130. n=8

S8=1.275

131. S5=15620words

132. S10=$1432.91

133. S20=1373(0dp)minutes

Page 26

134. a=$28656 S15=$642000 (3sf)135. a=1320 t10=2639136. tn=20 x (1.1)n–1

n=11 S11=371km137. r=0.794 a=5.04kg S8=20.6kg(3sf)

Page 28138.S∞=20139.S∞=6

140.S∞=50

Page 28 cont...

141.S∞= 1 31

189-b l

S∞=283.5

142.S∞=0.4

143.S∞=14.29(2dp)

144.S∞=56.89(2dp)

145.S∞=0.31

3( )146.S∞= .347 2 347 9

2o b l

147.r= 32 ,Seq.<9,6,4, 3

8 ,...>

148.r= 75- ,

Seq.<14,–10,7 71 , 5 49

5- ,...>

<14,–10,7.14,–5.10,...>

149.<28.8,–5.76,1.152,–0.2304,...>

Page 29

150.r=0.4,a=125,S∞=208.3

151.r=14,a=16384,S∞=21845. 3

152.r2= 94 ,r= 3

2!

,a=±54

whenr= 32 ,S∞=162

whenr= 32- ,S∞=

–32.4

153.r6= 40961

,r= 41!

,a=64

whenr= 41 ,S∞=85. 3

whenr= 41- ,S∞=51.2

154.a+ar=40 a(1+r)=40

72= 1 ra-] g

72= 1 140r r+ -] ]g g

1–r2=4072

r2= 94

r=±2

3

whenr= 32 ,a=24

whenr= 32- ,a=120

Page 21 cont...

102. 157837977=11 x 3n–1n=16

103. n=7

104. n=8

105. n=6

Page 22

106. 7r3=189

r=3Seq.<7,21,63,...>

107. 256r3=2048

r=2Seq.<64,128,256,...>

108. r=± 41 (±0.25)and

a=±81920

Seq.<–81920,–20480,–5120,..> Seq.<–81920,20480,–5120,..>

109. r=19 Seq.<129140163,

14348907,1594323,...>

110. r=2

3( .0 666o )

Seq.<9,6,4,...>

111. r=ab

Seq.<a,a2b,a3b2,...>

112. t12=4.00 x (1.15)12–1

t12=18.61km

113. t20=21.0 x (1.18)20–1

t20=487.5g

114. tn=850 x (0.95)n–1

100=850 x (0.95)n–1

n–1=41.7,n=42.7Answer=43.

115. Ifthewageincreasesby3%peryearthenthehourlyratemustincreaseby3%.A3%increaseisthesameasmultiplyingbyacommonratioof1.03.

Page 23116. Over21yearsagomeanshe

hashad20increasesattheconstantratioof1.03.

t1=16.00÷ (1.03)20

t1=$8.86perhour


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