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Arithmetic Sequences - Washingtonville Central School  · PDF file1 Arithmetic Sequences I know how to differentiate among arithmetic and geometric and I understand that sequences

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  • 1

    ArithmeticSequences

    IknowhowtodifferentiateamongarithmeticandgeometricandIunderstandthatsequencesandseriescanbeusedtomodelrealworldphenomena.

    BigIdea:Detectingpatternsinnumbershelpsstudentsseethemathematicalrelationshipsthatunderlierealworldphenomena.

  • 2

    GoalsandObjectivesStudentswillbeabletounderstandhowthecommondifferenceleadstothenexttermofanarithmeticsequence,theexplicitformforanArithmeticsequence,andhowtousetheexplicit

    formulatofindmissingdata.

    WhyDoWeNeedThis?Arithmeticsequencesareusedtomodelpatternsandformpredictionsforeventsbasedonthesepatterns,suchasin

    loanpayments,sales,andrevenue.

  • 3

    SEQUENCEDEFINITION:Asequenceisafunctionwhosesetofinputs,thedomain,isasubsetofthenaturalnumbers,i.e.{1,2,3,4,...}.

    Asequenceisoftenshownasanorderedlistofnumbers,calledthetermsorelementsofthesequence.Sequencefunctionnotationcanbetricky.

    AnArithmeticsequenceisthesetofnumbersfoundbyaddingthesamevaluetogetfromonetermtothenext.

    Vocabulary

    Example:

    1,3,5,7,...

    10,20,30,...

    10,5,0,5,..

  • 4

    Sequencesarefunctions.Thekeyhereisthattheinputissimplythenumbersplaceinlinesotospeakandtheoutputistheactualnumberinthelist.

  • 5

    Thecommondifferenceforanarithmeticsequenceisthevaluebeingaddedbetweenterms,andisrepresentedbythe

    variabled.

    Vocabulary

    Example:

    1,3,5,7,...d=2

    10,20,30,...d=10

    10,5,0,5,..d=5

  • 6

    Notation

    Aswestudysequencesweneedawayofnamingtheterms.Wewilluse:a1torepresentthefirstterm,a2torepresentthesecondterm,a3torepresentthethirdterm,andsooninthismanner.

    Ifweweretalkingaboutthe8thtermwewouldusea8.

    Whenwewanttotalkaboutgeneraltermcallitthenthtermandusean.

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    Solutio

    n

    1.Findtwosubsequenttermssuchasa1anda2

    2.Subtracta2a1

    FindingtheCommonDifference

    Findd:

    4,10,16,...

  • 8

    Findthecommondifference:

    1,4,7,10,...

    5,11,17,23,...

    9,5,1,3,...

    d=3d=6d=4d=21/2S

    olutions

  • 9

    NOTE:YoucanfindthecommondifferenceusingANYsetofconsecutiveterms

    Forthesequence10,4,2,8,...

    Findthecommondifferenceusinga1anda2:

    Findthecommondifferenceusinga3anda4:

    Whatdoyounotice?

  • 10

    Tofindthenextterm:

    1.Findthecommondifference

    2.Addthecommondifference(d)tothelasttermofthesequence

    3.Continueaddingforthespecifiednumberofterms

    Example:Findthenextthreeterms

    1,5,9,13,...

    d=95=4

    a5=13+4=17a6=17+4=21a7=21+4=25

    Solutio

    n

  • 11

    Findthenextthreeterms:

    1,4,7,10,...

    5,11,17,23,...

    9,5,1,3,...

    13,16,19

    29,35,41

    7,11,15

  • 12

    1 Findthenextterminthearithmeticsequence:3,9,15,21,...

    27

    Solutio

    n

  • 13

    2 Findthenextterminthearithmeticsequence:8,4,0,4,...

    8

    Solutio

    n

  • 14

    3 Findthevalueofdinthearithmeticsequence:10,2,14,26,...

    d=12

    Solutio

    n

  • 15

    4 Findthevalueofdinthearithmeticsequence:8,3,14,25,...

    d=11

    Solutio

    n

  • 16

    Writethefirstfourtermsofthearithmeticsequence

    thatisdescribed.

    1.Adddtoa12.Continuetoadddtoeachsubsequentterms

    Example:

    Writethefirstfourtermsofthesequence:

    a1=3,d=7 a1=3a2=3+7=10a3=10+7=17a4=17+7=24So

    lutio

    n

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    a1=4d=6

    a1=3d=3

    a1=0.5d=2.3

    a2=7d=5

    Findthefirstthreetermsforthearithmeticsequencedescribed:

    1.4,10,16,...

    2.3,0,3,...

    3..5,3.8,6.1,...

    4.7,12,17,...

    Solutio

    n

  • 18

    5 Whichsequencematchesthedescription?

    A 4,6,8,10B 2,6,10,14C 2,8,32,128

    D 4,8,16,32

    Solutio

    n

  • 19

    6 Whichsequencematchesthedescription?

    A 3,7,10,14B 4,7,10,13

    C 3,7,11,15

    D 3,1,5,9

    C

    Solutio

    n

  • 20

    OneofthemostfamousofallrecursivelydefinedsequencesisknownastheFibonacci

    Sequence.

    1,1,2,3,5,8,13,21,...

    wherethefirst2termsare1'sandanytermthereafteristhesumofprecedingtwo

    terms.Thisisasfamousasasequencecangetandisworth

    remembering.

  • 21

    RecursiveFormula

    Towritetherecursiveformulaforanarithmeticsequence:

    1.Finda12.Findd

    3.Writetherecursiveformula:

    Arecursiveformulaisonewhereeachterminthesequencedependsonatermortermsthatcamebeforeit.

  • 22

    Example:

    Writetherecursiveformulafor1,7,13,...

    a1=1d=71=6

    Solutio

    n

  • 23

    Writetherecursiveformulaforthefollowingsequences:

    a1=3d=3

    a1=0.5d=2.3

    1,4,7,10,...

    5,11,17,23,...

    Solutio

    n

  • 24

    7 Whichsequenceisdescribedbytherecursiveformula?

    A 2,8,16,...B 2,2,6,...C 2,6,10,...D 4,2,0,...

  • 25

    8 Arecursiveformulaiscalledrecursivebecauseitusesthepreviousterm.

    True False

  • 26

    9 Whichsequencematchestherecursiveformula?

    A 2.5,0,2.5,...

    B 5,7.5,9,...C 5,2.5,0,...D 5,12.5,31.25,...

  • 27

    ArithmeticSequence

    Tofindaspecificterm,saythe5thora5,youcouldwriteoutalloftheterms.

    Butwhataboutthe100thterm(ora100 )?

    Weneedtofindaformulatogettheredirectlywithoutwritingoutthewholelist.

    DISCUSS:Doesarecursiveformulahelpussolvethisproblem?

  • 28

    ArithmeticSequence

    Consider:3,9,15,21,27,33,39,...

    Doyouseeapatternthatrelatesthetermnumbertoits

    value?

    a1 3

    a2 9=3+6

    a3 15=3+12=3+2(6)

    a4 21=3+18=3+3(6)

    a5 27=3+24=3+4(6)

    a6 33=3+30=3+5(6)

    a7 39=3+36=3+6(6)

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    Thisformulaiscalledtheexplicitformula.

    Itiscalledexplicitbecauseitdoesnotdependonthepreviousterm

    Theexplicitformulaforanarithmeticsequenceis:

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    Tofindtheexplicitformula:1.Finda12.Findd

    3.Pluga1anddinto

    4.Simplify

    Example:Writetheexplicitformulafor4,1,6,...

    Solutio

    n

    a1=4

    d=14=5

    an=4+(n1)5

    an=45n+5

    an=95n

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    Writetheexplicitformulaforthesequences:

    1)3,9,15,...

    2)4,2.5,1,...

    3)2,0,2,...

    Solutio

    n

    1.an=3+(n1)6=3+6n6

    an=6n3

    2.an=4+(n1)2.5=4+2.5n2.5

    an=2.5n6.5

    3.an=2+(n1)(2)=22n+2

    an=42n

  • 32

    10 Theexplicitformulaforanarithmeticsequencerequiresknowledgeofthepreviousterm

    True False

    Solutio

    n

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    11 Writetheexplicitformulafor2,2,6,....

    A

    B

    C

    D Solution

  • 34

    12 Whichsequenceisdescribedby:

    A 7,9,11,...B 5,7,9,...C 5,3,1,...D 7,5,3,...

    Solutio

    n

  • 35

    FindingaSpecifiedTerm

    1.Findtheexplicitformulaforthesequence.

    2.Plugthenumberofthedesiredterminforn

    3.Evaluate

    Example:Findthe31sttermofthesequencedescribedby

    Solutio

    n n=31

    a31=3+2(31)

    a31=65

  • 36

    ExampleFindthe21sttermofthearithmeticsequencewitha1=4andd=3.

    an=a1+(n1)d

    a21=4+(211)3

    a21=4+(20)3

    a21=4+60

    a21=64

    Solutio

    n

  • 37

    ExampleFindthe12thtermofthearithmeticsequencewitha1=6andd=5.

    an=a1+(n1)d

    a12=6+(121)(5)

    a12=6+(11)(5)

    a12=6+55

    a12=49

    Solutio

    n

  • 38

    FindingtheInitialTermorCommonDifference

    1.Plugthegiveninformationintoan=a1+(n1)d

    2.Solvefora1,d,orn

    Example:Finda1forthesequencedescribedbya13=16andd=4

    an=a1+(n1)d

    16=a1+(131)(4)

    16=a1+(12)(4)

    16=a1+48

    a1=64

    Solutio

    n

  • 39

    ExampleFindthe1sttermofthearithmeticsequencewitha17=4andd=2.

    an=a1+(n1)d

    4=a1+(171)(2)

    4=a1+(16)(2)

    4=a1+32

    36=a1

    Solutio

    n

  • 40

    ExampleFinddofthearithmeticsequencewitha15=45anda1=3.

    an=a1+(n1)d

    45=3+(151)d

    45=3+(14)d

    42=14d

    3=d

    Solutio

    n

  • 41

    ExampleFindthetermnumbernofthearithmeticsequencewithan=6,a1=34andd=4.

    an=a1+(n1)d

    6=34+(n1)(4)

    6=34+4n4

    6=4n+38

    44=4n

    11=n

    Solutio

    n

  • 42

    13 Finda11whena1=13andd=6.

    an=a1+(n1)d

    a11=13+(111)(6)

    a11=13+(10)(6)

    a11=13+60

    a11=73

    Solutio

    n

  • 43

    GeometricSequences

  • 44

    GoalsandObjectivesStudentswillbeabletounderstandhowthecommonratioleadstothenexttermandtheexplicitformforanGeometricsequence,andusetheexplicitformulatofindmissingdata.

    WhyDoWeNeedThis?Geometricsequencesareusedtomodelpatternsandformpredictionsforeventsbasedonthesepatterns,suchasin

    loanpayments,sales,andrevenue.

  • 45

    Wehavestudiedtheideaofanarithmeticsequence,whereeachsuccessivenumberinthelistwasgeneratedbyaddingthesamequantitytothepreviousnumber.Recall:

    Exercise#1:Anarithmeticsequenceisdefinedrecursivelybythefollowingformula:

  • 46

    AnGeometricsequenceisthesetofnumbersfoundbymultiplyingbythesamevaluetogetfromonetermtothe

    next.

    Vocabulary

    Example:

    1,0.5,0.25,...

    2,4,8,16,...

    .2,.6,1.8,...

  • 47

    Theratiobetweeneveryconsecutiveterminthegeometricsequenceiscalledthecommonratio.

    Thisisthevalueeachtermismultipliedbytofindthenextterm.

    Vocabulary

    Example:

    1,0.5,0.25,...

    2,4,8,16,...

    0.2,0.6,1.8,...

    r=0.5

    r=2

    r=3

  • 48

    Writethefirstfourtermsofthegeometricsequencedescribed.

    1.Multiplya1bythecommonratior.

    2.Continuetomultiplybyrtofindeachsubsequentterm.

    Example:Findthefirstfourterms:

    a1=3andr=4a2=3*4=12

    a3=12*4=48

    a4