10/22/17

1

Unit 5 Recursive Functions

AFMMrs.Valentine

5.1Sequences

• Objective:Iwillbeabletoidentifyaninfiniteseries.Iwillbeabletoreadandwriterecursionformulas.Iwillbeabletographsequences.

• VocabularyFibonaccisequence

GeneralTerm RecursionFormulas

FiniteSequence Graphofasequence

InfiniteSequence

5.1Sequences

•  Sequences• Manycreationsinnatureinvolveintricate

mathematicaldesigns,includingavarietyofspirals.• TheFibonaccisequencecanbefoundinnaturein

manyplaces.•  Thesequencebeginswithtwoonesandeveryterm

thereafteristhesumofthetwoprecedingterms•  1,1,2,3,5,8,13,21,24,55,89,144,233,…an

•  anisageneraltermofthesequence,alsocalledthenthterm.

5.1Sequences

5.1Sequences

• Aninfinitesequenceisafunctionwhosedomainisthesetofpositiveintegers.Thefunctionvalues,orterms,ofthesequencearerepresentedby

a1,a2,a3,a4,…,an,…

Sequenceswhosedomainsconsistonlyofthefirstnpositiveintegersarecalledfinitesequences.

• Becauseasequenceisafunctionwhosedomainisthesetofpositiveintegers,thegraphofasequenceisasetofdiscretepoints(notconnected).

5.1Sequences

10/22/17

2

5.1Sequences

• RecursionFormulas• Arecursionformuladefinesthenthtermofa

sequenceasafunctionofthepreviousterm.• Example:an=3an-1+2• Recursionformulascanbeusedtofindthenextterm

inasequence,butyoumustalreadyknowthepreviousterminordertofindit.

5.2SummaryNotation

• Objective:Iwillbeabletoread,write,andevaluatefactorialnotation.Iwillbeabletoread,write,andmanipulatesummationnotationforthesumsofasequence.

• Vocabulary

FactorialNota@on

Summa@onNota@on

IndexofSumma@on

UpperLimitofSumma@on

LowerLimitofSumma@on

ExpandingtheSumma@onNota@on

5.2SummaryNotation

•  FactorialNotation• Productsofconsecutivepositiveintegersoccurquite

ofteninsequences.•  Factorialnotationisaspecialnotationforconsecutive

products.•  Ifnisapositiveinteger,thenotationn!(Read“n

factorial)istheproductofallpositiveintegersfromndownthrough1.

n!=n(n-1)(n-2)…(3)(2)(1)•  0!=1bydefinition

5.2SummaryNotation

•  Likeexponents,factorialsonlyaffectthenumbertheyaredirectlyfollowingunlessgroupingsymbols,likeparentheses,appear.•  2*3!=2(3*2*1)=12•  (2*3)!=6!=6*5*4*3*2*1=720

5.2SummaryNotation

•  Itcanbeusefultofindthesumofthefirstntermsofasequence.

•  Iftherearetoomanyterms,writingouttheentiresumcanbecumbersome.

• Summationnotationcanbeusedtosimplifywritingsumsofasequence.•  Thesumofthefirstntermsofasequenceis

representedby

5.2SummaryNotation

•  iistheindexofsummation,whichindicatesthedomainofthetermsofthesequencebeingadded.

•  nistheupperlimitofsummation(thelargestvalueoftheindex)

•  1isthelowerlimitofthesummation(thesmallestvalueoftheindex)

10/22/17

3

5.2SummaryNotation

•  NOTES:•  Anylettercanbeusedfortheindex•  Thelowerlimitcanbeothernumbersthan1

•  Writingoutthesumofthetermsiscalledexpandingthesummationnotation

5.2SummaryNotation

• PropertiesofSums

5.3ArithmeticSequences

• Objective:Iwillbeabletorecognizeanarithmeticsequenceandgraphitsfunction.Iwillbeabletofindthecommondifferencefroanarithmeticsequenceandgiveasetofvaluesforthesequence.

• Vocabulary

Arithme@cSequence

CommonDifference

GeneralTermofArithme@cSequence

5.3ArithmeticSequences

•  ArithmeticSequences• Anarithmeticsequenceisasequenceinwhicheach

termafterthefirstdiffersfromtheprecedingtermbyaconstantamount.

• Thedifferencebetweenconsecutivetermsiscalledthecommondifferenceofthesequence.

5.3ArithmeticSequences

•  FindingtheCommonDifferenceforanArithmeticSequence• Representedbythesymbold.• Subtractanytwoconsecutivepointsinanarithmetic

sequencetofindthecommondifference.

d=an-an-1

5.3ArithmeticSequences

• Anarithmeticsequenceisalinearfunctionwhosedomainisthesetofpositiveintegers•  Thegraphofthisfunctionisasetofdiscretepoints

(pointsnotconnected)•  Thepointslieonastraightline.

• WritingtheTermsofanArithmeticSequence•  Towriteasetoftermsinanarithmeticsequence,you

mustknowatleastoneterm,usuallya1,andthecommondifference.

• Usetherecursionformulaan=an-1+d

10/22/17

4

5.4ArithmeticSums

• Objective:Iwillbeabletofindaspecifictermwithinanarithmeticsequence.Iwillbeabletofindthesumofthefirstntermsofanarithmeticsequence.

• VocabularySumoftheFirstnTermsofArithme@cSequence

5.4ArithmeticSums

•  TheGeneraltermofanArithmeticSequence•  Considerhowthefirstsixtermsinanarithmetic

sequencearedetermined:

• Thenthterm(generalterm)ofanarithmeticsequencewiththefirstterma1andcommondifferencedis

an=a1+(n-1)d

5.4ArithmeticSums

•  TheSumoftheFirstnTermsofanArithmeticSequence• Thesumofthefirstntermsofanarithmetic

sequence,denotedbySnandcalledthenthpartialsum,canbefoundwithouthavingtoaddupalltheterms.

• Thesum,Snofthefirstntermsofanarithmeticsequenceisgivenby

5.5GeometricSequences

• Objective:Iwillbeabletorecognizegeometricsequences.Iwillbeabletofindthecommonratioandasetoftermsforageometricsequence.Iwillbeabletofindthegeneraltermofageometricsequence.

• Vocabulary

GeometricSequence

CommonRa@o GeneralTermofaGeometricSequence

5.5GeometricSequences

• GeometricSequences• Ageometricsequenceisasequenceinwhicheach

termafterthefirstisobtainedbymultiplyingtheprecedingtermbyafixednonzeroconstant.

• Theamountbywhichwemultiplyeachtermiscalledthecommonratioofthesequence.

5.5GeometricSequences

•  FindtheCommonRatioofaGeometricSequence• Thecommonratiorisfoundbydividinganyterm

afterthefirsttermbythetermthatdirectlyprecedesit.

• Ageometricsequencewithapositivecommonratiootherthan1isanexponentialfunctionwhosedomainisthepositivesetofintegers.

10/22/17

5

5.5GeometricSequences

• WriteTermsofaGeometricSequence• Towriteasetoftermsinageometricsequence,you

mustknowatleastoneterm,usuallya1,aswellasthecommonratio.

• Usetherecursionformula:an=an-1*r

•  TheGeneralTermofaGeometricSequence• Thenthterm(thegeneralterm)ofageometric

sequencewiththefirstterma1andthecommonratioris

5.6GeometricSeries

• Objective:Iwillbeabletofindthesumofntermsofageometricsequence.Iwillbeabletounderstandandfindvaluesofanannuity.Iwillbeabletousegeometricseriesandtheirsumstofindmultipliereffectvaluesontheeconomy.

• VocabularynthPar@alSum Annuity ValueoftheAnnuity Mul@plier

EffectInfiniteGeometricSeries SumofInfiniteGeometricSeries

5.6GeometricSeries

•  TheSumoftheFirstnTermsofaGeometricSequence• Thesumofthefirstntermsofageometricsequence,

denotedbySnandcalledthenthpartialsum,canbefoundwithouthavingtoaddupalltheterms.

• Thesumofthefirstntermsofageometricsequenceisgivenby

•  Thecommonratiocannotequal1.

5.6GeometricSeries

• Annuities• Anannuityisasequenceofequalpaymentsmadeat

equaltimeperiods.• AnIRAisanexampleofanannuity.• Thetermsinthissequencearebasedonthesimple

interestformula:A=P1+rt• Thevalueoftheannuityisthesumofalldeposits

madeplusallinterestpaid.

5.6GeometricSeries

•  IfPisthedepositmadeattheendofeachcompoundingperiodforanannuityatrpercentannualinterestcompoundedntimesperyear,thevalue,A,oftheannuityaftertyearsis

5.6GeometricSeries

• GeometricSeries• Aninfinitesumintheform

withthefirstterma1andcommonratioriscalledaninfinitegeometricseries.

10/22/17

6

5.6GeometricSeries

• ThesumofanInfiniteGeometricSeries•  If-1<r<1,thenthesumoftheinfinitegeometric

seriesaboveisgivenby

•  If|r|≥1,theinfiniteseriesdoesnothaveasum.

5.6GeometricSeries

• MultiplierEffect• Ataxrebatethatreturnsacertainamountof

moneytotaxpayerscanhaveatotaleffectontheeconomythatismanytimesthisamount.

•  Ineconomics,thisphenomenoniscalledthemultipliereffect.

•  Thiscanbefoundbyfindingthesumofaninfinitegeometricseries.