Unit 6 SequencesMrs.Valen+ne

CCM3

6.1 Sequences and Series•  Genera&ngaSequenceUsinganExplicitFormula

–  Sequence:orderedlistofnumbers(eachoneisaterm)•  Termsaresymbolizedwithavariableandasubscriptlabel(ex:a1,a2,etc.)

•  Thesubscriptsindicatetheorderoftheterms.–  Explicitformula:describesthenthtermofasequenceusingthenumbern.•  Sequence:2,4,6,8,10àthenthtermistwicethevalueofnsotheformulaisan=2n.

•  Example:asequencehastheexplicitformulaan=3n–2.Whatarethefirsttentermsofthissequence?

3(1)–2=13(2)–2=43(3)–2=7

3(4)–2=103(5)–2=133(6)–2=16

3(7)–2=193(8)–2=223(9)–2=253(10)–2=28

Thefirsttentermsare:1,4,7,10,13,16,19,22,25,28

6.1 Sequences and Series•  Wri&ngaRecursiveDefini&onforaSequence

–  Recursivedefini+onhastwoparts:•  Ini+alcondi+on(ex:a1=133)•  Recursiveformula(rela+ngeachtermtotheonebeforeit):an=an-1–3forn>1

–  Example:thenumberofblocksinatwo-dimensionalpyramidisasequencethatfollowsarecursiveformula.Whatistherecursivedefini+on?

136101521

a2–a1=3–1=2a3–a2=6–3=3a4–a3=10–6=4a5–a4=15–10=5a6–a5=21–15=6an–an–1=n

a1=1andan=an–1+n

Hint:Lookforsimpleaddi+on/mul+plica+onpa^ernstorelateconsecu+veterms.

6.1 Sequences and Series•  Wri&nganExplicitFormulaforaSequence

–  Recursiveformulasworkforsmallsec+onsofsequences–  Explicitformulasarebe^erforlargersec+ons.–  Example:Whatisthe100thtermofthepyramidsequence?

a1 a2 a3 a4 a5 … an

1 1+2 1+2+3 1+2+3+4 1+2+3+4+5 … 1+2+…+n

1 3 6 10 15 … ____

an=1+2+3+…+(n–2)+(n–1)+n an=n+(n–1)+(n–2)+…+3+2+1or

an=1+2+3+…+(n–2)+(n–1)+nan=n+(n–1)+(n–2)+…+3+2+1

2an=(n+1)+(n+1)+(n+1)+…+(n+1)+(n+1)+(n+1)2an=n(n+1)an=½n(n+1)

an=½n(n+1)a100=½(100)(100+1)a100=(50)(101)a100=5050

+

6.1 Sequences and Series•  UsingFormulastoFindTermsofaSequence

–  Pierrebegantheyearwithanunpaidbalanceof$300onhiscreditcard.Becausehehadnotreadthecreditcardagreement,hedidnotrealizethatthecompanycharged1.8%interesteachmonthonhisunpaidbalance,inaddi+ontoa$29penaltyinanymonthhemightfailtomakeaminimumpayment.Pierreignoredhiscreditcardbillfor4consecu+vemonthsbeforefinallydecidingtopayoffthebalance.Whatdidheoweager4monthsofnon-payment?

a0=300

Recursiveformula:an=1.018*an-1+29

IntheMODEmenuofyourcalculator,selecttofloat2decimalplaces.

Ager4months,Pierreowes$441.36

6.2 Arithmetic Series•  Iden&fyingArithme&cSequences

–  Arithme+csequences:differencebetweenanytwoconsecu+vetermsisalwaysthesamenumber.

–  Thisdifferenceiscalledcommondifference(d).

–  Example:Isthesequence3,6,9,12,15,…anarithme+csequence?

–  Isthesequence1,4,9,16,25,…anarithme+csequence?

a,a+d,a+2d,a+3d,…Recursivedef.:a1=aandan=an–1+dforn>1Explicitdef.:an=a+(n–1)dforn≥1

3691215

3333

Thedifferencebetweeneachtwotermsis3,soithasacommondifference(d=3).Itisanarithme+csequence.

No,thereisnocommondifference.

6.2 Arithmetic Series•  AnalyzingArithme&cSequences

–  Whatisthe100thtermofthearithme+csequencethatbegins6,11,…?

–  Whatarethesecondandthirdtermsofthearithme+csequence100,__,___,82,...?

11–6=5=dan=a+(n–1)da100=6+(100–1)5a100=501

82=100+3d–18=3d–6=d

100–6=9494–6=88

100,94,88,82,…

Thesecondandthirdtermsare94and88,respec+vely.

6.2 Arithmetic Series•  UsingArithme&cMean

–  Thearithme+cmeanoftwonumbersis–  Inanarithme+csequence,themiddletermofanythreeconsecu+vetermsisthearithme+cmeanoftheothertwoterms.

–  Example:Whatisthemissingtermofthearithme+csequence…,15,____,59,...?

–  The9thand11thtermsofanarithme+csequenceare132and98.Whatisthe10thterm?

6.2 Arithmetic Series•  UsinganExplicitFormulaforanArithme&cSequence

–  Thenumberofseatsinthefirst13rowsinasec+onofanarenaformanarithme+csequence.Rows1and2areshowninthediagram.HowmanyseatsareinRow13?

–  Supposeatrolleystopsatacertainintersec+onevery14min.Thefirsttrolleyofthedaygetstothestopat6:43am.Howlongdoyouhavetowaitforatrolleyifyougettothestopat8:15am?

16–14=2=dan=a1+(n–1)da13=14+(13–1)2a13=38

a1=43,d=14an=43+(n–1)14135=43+14n–147.57=nàneedn=8

6:43ais43minpast6a8:15ais135minpast6a

a1=43,d=14a8=43+(8–1)14a8=141141–135=6minwait

Row2:16seats

Row1:14seats

6.2 Arithmetic Series•  FindingtheSumofaFiniteArithme&cSeries

–  Seriesistheindicatedsumofthetermsofasequence.•  Finiteserieshaveafirstandlastterm•  Infiniteseriescon+nuewithoutend

–  Anarithme+cseriesisaserieswhosetermsformanarithme+csequence.

–  ThesumSnofafinitearithme+cseriesa1+a2+a3+…+anis

6.2 Arithmetic Series–  Arithme+cseriescanusesubtrac+ontofindthetotalnumberofterms.Recallthatthecommondifference=d.

–  Example:Whatisthesumoftheevenintegersfrom2to100?

–  Whatisthesumofthefinitearithme+cseries4+9+14+19+24+…+99

Arithme+cSeries:2+4+6+…+100a1=2anda50=100

S50=25(102)=2550

9-4=5(commondiff)99-4=95/5=19+1=20terms S50=10(103)=1030

6.2 Arithmetic Series•  UsingtheSumofFiniteArithme&cSeries

–  Acompanypays$10000bonustosalespeopleattheendoftheirfirst50weeksiftheymake10salesintheirfirstweek,andthenimprovetheirsalesnumbersbytwoeachweekthereager.Onesalespersonqualifiedforthebonuswiththeminimumpossiblenumberofsales.Howmanysalesdidthesalespersonmakeinweek50?Inall50weeks?

a1=10d=2an=a1+(n–1)da50=10+(50–1)(2)a50=108sales S50=2950sales

6.2 Arithmetic Series•  Wri&ngaSeriesinSumma&onNota&on

–  TheGreekcapitalle^ersigma,Σ,isusedtoindicateasum.Withit,youuselimitstoindicatehowmanytermsyouareadding.•  Limitsaretheleastandgreatestvaluesofnintheseries.Youwritethelimitsaboveandbelowthecapitalsigma.

•  32+42+52+…+1082canbewri^enas

•  Foraninfiniteseries,theupperlimitiswri^enasinfinity(∞).•  Tofindthenumberoftermsinaseries,subtractthelowerlimitfromtheupperlimitandadd1.

108–3+1=106terms

6.2 Arithmetic Series–  Example:whatisthesumma+onnota+onfortheseries?7+11+15+…+203+207

–  Whatisthesumma+onnota+onfortheseries–5+2+9+16+...+261+268

an=a1+(n–1)dan=7+(n–1)(4)an=7+4n–4an=4n+3

an=4n+3207=4n+3204=4n51=n

a1=7d=4

a1=–5d=7

an=a1+(n–1)dan=–5+(n–1)(7)an=–5+7n–7an=7n–12

an=7n–12268=7n–12280=7n40=n

6.2 Arithmetic Series•  FindingtheSumofaSeries

–  Iftheexplicitformulaforthenthterminsumma+onnota+onisalinearfunc+onofn,thentheseriesofarithme+c.Theslopeofthelinearfunc+onisthecommondifferencebetweentermsoftheseries.

–  Example:whatisthesumoftheserieswri^eninsumma+onnota+on?

a1=5(1)+3=8a70=5(70)+3=353

S70=35(361)S70=12635

6.2 Arithmetic Series•  UsingaGraphingCalculatortoFindtheSumofaSeries

–  Onagraphingcalculator,youcanfindthesumofafiniteseriesbyusingcommandsfromtheLISTmenu.

–  Example:Whatisthesumoftheserieswri^eninsumma+onnota+on?

Thesumis12635

6.3 Geometric Series•  Iden&fyingGeometricSequences

–  Geometricsequencehasstar+ngvalueaandcommonra+ora,ar,ar2,ar3,…

•  Recursivedefini+on:a1=aandan=a1�rforn>1•  Explicitdefini+on:an=a1�rn-1forn≥1

–  Example:Isthesequencegeometric?Ifitis,whatarea1andr?3,6,12,24,48

35,310,315,320,…

6/312/624/1248/24

36122448 Allfrac+onsreducedownto2.

Thisisageometricsequence.a1=3andr=2

310/35315/310320/315

35310315320Allfrac+onsreducedownto35.

Thisisageometricsequence.a1=35andr=35

6.3 Geometric Series•  AnalyzingGeometricSequences

–  Whatisthe10thtermofthegeometricsequence4,12,36,…?

–  Findthesecondandthridtermsinthegeometricsequence2,__,__,–54,...

r=12/4=3a1=4

an=a1�rn–1

a10=4�(3)10–1a10=78732

a1=2a4=–54

a4=a1�r4–1–54=2�r3–27=r3–3=r

a2=2�(–3)=–6a3=–6�(–3)=18

Thesecondandthirdtermsare–6and18,respec+vely.

6.3 Geometric Series•  UsingaGeometricSequence

–  Whenaballbounces,theheightsofconsecu+vebouncesformageometricsequence.Whataretheheightsofthe4thand5thbounces?

a1=100a3=49

a3=a1�r3–149=100�r249/100=r2√(49/100)=r7/10=r

a4=49�(7/10)=34.3a5=34.3�(7/10)=24

Thefourthandfighbouncesare34.3cmand24cm,respec+vely.

6.3 Geometric Series•  UsingtheGeometricMean

–  Thesquareofthemiddletermofanythreeconsecu+vetermsisequaltotheproductoftheothertwoterms.

–  Geometricmeanofxandy:•  Thegeometricmeanisposi+vebydefini+on.•  Therefore,therearetwopossibili+esforamissingterm:theposi+veandthenega+veversionofthemean.

–  Example:Whatarethepossiblevaluesofthemissingtermofthegeometricsequence48,__,3,…?

Thepossiblemissingvaluesforthesequenceare±12.

6.3 Geometric Series•  FindingtheSumsofFiniteGeometricSeries

–  Youcanwriteanywholenumberthathasthesamedigitineveryplaceasthesumofthetermsofageometricsequence.

4444=4(100)+4(101)+4(102)+4(103)•  Therefore,anyra+onalnumbercanbewri^enasageometricsequence.47/90=0.5222…=0.5+2(0.1)2+2(0.1)3+2(0.1)4+...

–  GeometricSeries:sumoftermsinageometricsequence–  Thesumofageometricseries,Sn,whenr≠1,is

6.3 Geometric Series–  Example:Whatisthesumofthefinitegeometricseries3+6+12+24+…+3072?

–  Evaluate

a1=3r=6/3=2

3072=3�2n–11024=2n–1210=2n–110=n-111=n

a1=4�(0.5)0=4

≈8

6.3 Geometric Series•  UsingtheGeometricSeriesFormula

–  Afamousstoryinvolvesasoldierwhorescueshiskinginba^le.Thekinggrantshimanyprize“withinreason”fromtherichesofthekingdom.Thesoldierasksforachessboardwithasinglekernelofwheatonthefirstsquare,twokernelsofwheatonthesecondsquare,thenfour,theneight,andsoonforall64squaresofthechessboard.Thekingdecidesthattherequestisreasonable.Howmanytotalkernelsofwheatdidthesoldierrequest?

a1=1r=4/2=2n=64

=264–1≈1.845x1019

Thesoldierrequested1.845x1019kernelsofwheat.

6.3 Geometric Series•  AnalyzingInfiniteGeometricSeries

–  Thetermsofageometricseriesgrowrapidly(r>1)ordiminishrapidly(0<r<1).

–  When|r|<1,theseriesdiminishessorapidlythataninfinitegeometricserieshasafinitesum.

•  TheseriesconvergestoSasngetsverylargeif|r|<1•  Theseriesdivergesifitdoesnotconvergetoasum(|r|≥1)•  Example:Doestheseriesconvergeordiverge?Ifitconverges,whatisthesum?1+½+¼+…

r=½÷1=½|r|=|½|<1 Theseriesconverges