# 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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ArithmeticSequences

IknowhowtodifferentiateamongarithmeticandgeometricandIunderstandthatsequencesandseriescanbeusedtomodelrealworldphenomena.

BigIdea:Detectingpatternsinnumbershelpsstudentsseethemathematicalrelationshipsthatunderlierealworldphenomena.

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formulatofindmissingdata.

WhyDoWeNeedThis?Arithmeticsequencesareusedtomodelpatternsandformpredictionsforeventsbasedonthesepatterns,suchasin

loanpayments,sales,andrevenue.

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

Asequenceisoftenshownasanorderedlistofnumbers,calledthetermsorelementsofthesequence.Sequencefunctionnotationcanbetricky.

Vocabulary

Example:

1,3,5,7,...

10,20,30,...

10,5,0,5,..

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Sequencesarefunctions.Thekeyhereisthattheinputissimplythenumbersplaceinlinesotospeakandtheoutputistheactualnumberinthelist.

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variabled.

Vocabulary

Example:

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

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

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

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Notation

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

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Solutio

n

1.Findtwosubsequenttermssuchasa1anda2

2.Subtracta2a1

FindingtheCommonDifference

Findd:

4,10,16,...

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Findthecommondifference:

1,4,7,10,...

5,11,17,23,...

9,5,1,3,...

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

olutions

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NOTE:YoucanfindthecommondifferenceusingANYsetofconsecutiveterms

Forthesequence10,4,2,8,...

Findthecommondifferenceusinga1anda2:

Findthecommondifferenceusinga3anda4:

Whatdoyounotice?

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Tofindthenextterm:

1.Findthecommondifference

Example:Findthenextthreeterms

1,5,9,13,...

d=95=4

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

Solutio

n

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Findthenextthreeterms:

1,4,7,10,...

5,11,17,23,...

9,5,1,3,...

13,16,19

29,35,41

7,11,15

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

27

Solutio

n

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

8

Solutio

n

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

d=12

Solutio

n

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

d=11

Solutio

n

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Writethefirstfourtermsofthearithmeticsequence

thatisdescribed.

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

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5 Whichsequencematchesthedescription?

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

D 4,8,16,32

Solutio

n

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6 Whichsequencematchesthedescription?

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

C 3,7,11,15

D 3,1,5,9

C

Solutio

n

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OneofthemostfamousofallrecursivelydefinedsequencesisknownastheFibonacci

Sequence.

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

wherethefirst2termsare1'sandanytermthereafteristhesumofprecedingtwo

terms.Thisisasfamousasasequencecangetandisworth

remembering.

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RecursiveFormula

Towritetherecursiveformulaforanarithmeticsequence:

1.Finda12.Findd

3.Writetherecursiveformula:

Arecursiveformulaisonewhereeachterminthesequencedependsonatermortermsthatcamebeforeit.

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Example:

Writetherecursiveformulafor1,7,13,...

a1=1d=71=6

Solutio

n

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Writetherecursiveformulaforthefollowingsequences:

a1=3d=3

a1=0.5d=2.3

1,4,7,10,...

5,11,17,23,...

Solutio

n

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7 Whichsequenceisdescribedbytherecursiveformula?

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

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8 Arecursiveformulaiscalledrecursivebecauseitusesthepreviousterm.

True False

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9 Whichsequencematchestherecursiveformula?

A 2.5,0,2.5,...

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

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ArithmeticSequence

Tofindaspecificterm,saythe5thora5,youcouldwriteoutalloftheterms.

Weneedtofindaformulatogettheredirectlywithoutwritingoutthewholelist.

DISCUSS:Doesarecursiveformulahelpussolvethisproblem?

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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

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10 Theexplicitformulaforanarithmeticsequencerequiresknowledgeofthepreviousterm

True False

Solutio

n

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11 Writetheexplicitformulafor2,2,6,....

A

B

C

D Solution

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12 Whichsequenceisdescribedby:

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

Solutio

n

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FindingaSpecifiedTerm

1.Findtheexplicitformulaforthesequence.

2.Plugthenumberofthedesiredterminforn

3.Evaluate

Example:Findthe31sttermofthesequencedescribedby

Solutio

n n=31

a31=3+2(31)

a31=65

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ExampleFindthe21sttermofthearithmeticsequencewitha1=4andd=3.

an=a1+(n1)d

a21=4+(211)3

a21=4+(20)3

a21=4+60

a21=64

Solutio

n

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ExampleFindthe12thtermofthearithmeticsequencewitha1=6andd=5.

an=a1+(n1)d

a12=6+(121)(5)

a12=6+(11)(5)

a12=6+55

a12=49

Solutio

n

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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

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ExampleFindthe1sttermofthearithmeticsequencewitha17=4andd=2.

an=a1+(n1)d

4=a1+(171)(2)

4=a1+(16)(2)

4=a1+32

36=a1

Solutio

n

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ExampleFinddofthearithmeticsequencewitha15=45anda1=3.

an=a1+(n1)d

45=3+(151)d

45=3+(14)d

42=14d

3=d

Solutio

n

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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

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13 Finda11whena1=13andd=6.

an=a1+(n1)d

a11=13+(111)(6)

a11=13+(10)(6)

a11=13+60

a11=73

Solutio

n

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GeometricSequences

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WhyDoWeNeedThis?Geometricsequencesareusedtomodelpatternsandformpredictionsforeventsbasedonthesepatterns,suchasin

loanpayments,sales,andrevenue.

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Exercise#1:Anarithmeticsequenceisdefinedrecursivelybythefollowingformula:

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AnGeometricsequenceisthesetofnumbersfoundbymultiplyingbythesamevaluetogetfromonetermtothe

next.

Vocabulary

Example:

1,0.5,0.25,...

2,4,8,16,...

.2,.6,1.8,...

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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

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Writethefirstfourtermsofthegeometricsequencedescribed.

1.Multiplya1bythecommonratior.

2.Continuetomultiplybyrtofindeachsubsequentterm.

Example:Findthefirstfourterms:

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

a3=12*4=48

a4

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