Download pdf - Arithmetic and Geometric Sequences 3.1 Set, Go!msshultis.weebly.com/uploads/1/0/9/3/10930910/2016-17_sduhsd_math_1_cp... · Name: Arithmetic and Geometric Sequences 3.2 Ready, Set,

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SDUHSDMath1CollegePrep

Name: ArithmeticandGeometricSequences 3.1Ready,Set,Go!ReadyTopic:Exponents,substitution,andfunctionnotationFindeachvalue.1. 3 2. 3 3. 3 4. 3

3 9 27 81Foreachofthefollowing,find , and 5. 2 6. 3 , , , , 7. 2 1 3

, , Completeeachtable.8.

Term 1st 2nd 3rd 4th 5th 6th 7th 8th

Value 2 4 8 16 32 64 128 256

9.


Value 66 50 34 18 2

10.


Value 3 9 27 81 729 6561

11.


Value 160 80 40 20 10 5 2.5 1.25

12.


Value 9 2 5 12 19 26 33 40

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SetTopic:CompletingatableFillinthetable.Thenwriteasentenceexplaininghowyoufiguredoutthevaluestoputineachcell.Explainhowtofigureoutwhatwillbeincell#8.13.Yourunabusinessmakingbirdhouses.Youspend$600tostartyourbusiness,anditcostsyou$5.00

tomakeeachbirdhouse.

#ofbirdhouses 0 1 2 3 4 5 6

Totalcosttobuild 600 605 610 615 620 625 630

Explanation:Forthefirstbirdhouseitcosts$600tostartthebusiness,plus$5forthefirstbirdhouse,andafterthatitisjust$5moreforeachadditionalbirdhouse.Thecostformaking8birdhousesisthen$ ⋅ ,or$640.

14.Youborrow$500fromarelative,andyouagreetopaybackthedebtatarateof$15permonth.

#ofmonths 1 2 3 4 5 6 7

Amountofmoneyowed 500 485 470 455 440 425 410

Explanation:Onmonth#1youowethetotal,$500,theneverymonthafterthatyouowe$15lessbecauseyoupaid$15towardthedebtattheendofeachmonth.Onthe8thmonth,theamountofmoneyowedwouldbe ,or$395

15.Youearn$10perweek.

#ofweeks 1 2 3 4 5 6 7

Amountofmoneyearned 10 20 30 40 50 60 70

Explanation:Themoneyearnedis$10everyweek,soadd$10tothetotalearnedeachweek.Onthe8thweektheamountofmoneyearnedwouldthenbe ,or$80.

16.Youaresavingforabikeandcansave$10perweek.Youhave$25alreadysaved.

#ofweeks 0 1 2 3 4 5 6

Amountofmoneysaved 25 35 45 55 65 75 85

Explanation:Onweek1youhavethe$25youalreadysavedplusthe$10forthatweek,so$35.Foreveryweekafterthatadd$10tothemoneysavedfortheadditional$10savedthatweek.Onthe8thweekthetotalamountofmoneysavedis ,or$105.

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GoTopic:GoodviewingwindowWhensketchingagraphofafunction,itisimportantthatweseekeypoints.Forlinearfunctions,wewantawindowthatshowsimportantinformationrelatedtothestory.Often,thismeansincludingboththex‐andy‐interceptsFindanappropriategraphingwindowforeachofthefollowinglinearfunctions.Sketchthegraph.Fillintheblanksshowingtheloweranduppervaluesandincludethescaleforeachaxis.YoumayuseanonlinegraphingutilitysuchasDesmos(https://www.desmos.com/calculator)orMATHPAPA(https://www.mathpapa.com/calc.html?q=)Answersmayvary.Sampleanswersprovidedbelow:17. 1

x: , byy: , x‐scale: y‐scale:

18. 7 3 14x: , byy: , x‐scale:1 y‐scale:1



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Name: ArithmeticandGeometricSequences 3.2Ready,Set,Go!ReadyTopic:Findingvaluesforapattern1. BobCooperwasbornin1900.By1930hehad3sons,allwiththeCooperlastname.By1960eachof

Bob’s3boyshadexactly3sonsoftheirown.Bytheendofeach30yeartimeperiod,thepatternofeachCooperboyhavingexactly3sonsoftheirowncontinued.HowmanyCoopersonswereborninthe30yearperiodbetween1960and1990?27

2. Createadiagramthatwouldshowthispattern.

Year 1900 1930 1960 1990 2020#ofsons 0 3 9 27 81

3. PredicthowmanyCoopersonswillbebornbetween1990and2020,ifthepatterncontinues. 814. TrytowriteanequationthatwouldhelpyoupredictthenumberofCoopersonsthatwouldbeborn

between2020and2050.Ifyoucan’tfindtheequation,explainitinwords.

SetTopic:EvaluatingequationsEvaluatethefollowingequationswhen , , , , .Organizeyourinputsandoutputsintoatableofvaluesforeachequation.Letxbetheinputandybetheoutput.5. 4 6. 3 7. 3 8. 10

x y1 42 163 644 2565 1024

x y12 934 815

x y12345

x y1 102 1003 10004 100005 100000

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GoTopic:SolveequationsSolvethefollowingequationsfortheunknownvariable.Checkyouranswer.9. 3 1 2 3

10. 7 20 5

.

11.9 2 3 3

.

12.2

13.3 3 2 1 0

14. 6 3 5 3 2 2 2 9

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Name: ArithmeticandGeometricSequences 3.3Ready,Set,Go!ReadyTopic:Writetheequationofalinegiventwopoints.Grapheachpairofpoints,drawalinethatgoesthroughbothpoints,andwriteanequationofthatline.1. 5, 2 and 7, 0 2. 4, 2 and 6, 7

Equation: Equation: 3. 3, 0 and 0, 4 4. 2, 4 and 2, 6

Equation: Equation: 5. Writetheequationofthelinethatpassesthroughthepoints 2, 2 and 8, 8 withoutthehelpofagraph.

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SetTopic:RecursiveandexplicitfunctionsofarithmeticsequencesBelowyouaregivenvarioustypesofinformation.Writetherecursiveandexplicitfunctionsforeacharithmeticsequence.Finally,grapheachsequence,makingsureyouclearlylabelyouraxes.6. 2, 4, 6, 8, …

Recursive:

, Explicit:

7.

Recursive:

, Explicit:

Time(Days)

#ofcells

1 32 63 94 12

8. Clairehas$300inanaccount.Shedecidessheisgoingtotakeout$25eachmonth.

Recursive:

, Explicit:

9. EachdayTaniadecidestodosomethingnicefor2strangers.WriterecursiveandexplicitequationsthatrepresentthenumberofstrangersTaniathatdoessomethingniceforeachday(nottotalnumberofstrangers).

Recursive:

, Explicit:

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10.

Recursive:

, Explicit:

GoTopic:RecursiveandexplicitfunctionsofgeometricsequencesBelowyouaregivenvarioustypesofinformation.Writetherecursiveandexplicitfunctionsforeachgeometricsequence.Finally,grapheachsequence,makingsureyouclearlylabelyouraxes.11.2, 4, 8, 16, …

Recursive:

, Explicit:

12.

Recursive:

, Explicit:

Time(Days)

#ofcells

1 36

3 124 24

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13.Clairehas$300inanaccount.Shedecidessheisgoingtotakeouthalfofwhat’sleftinthereattheendofeachmonth.

Recursive:

, Explicit:

14. Taniacreatesachainletterandsendsittofourfriends.Eachdayeachfriendistheninstructedtosendittofourfriendsandsoforth.

Recursive:

, Explicit:

15.

Recursive:

, Explicit:

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Name: ArithmeticandGeometricSequences 3.4Set,Go!SetTopic:DeterminerecursiveequationsTwoconsecutivetermsinanarithmeticsequencearegiven.Findtheconstantdifferenceandtherecursiveequation.1. If 3 5and 4 8.

5 , 6 RecursiveFunction: , 2. If 2 20and 3 12.

4 , 5 RecursiveFunction: , 3. If 5 3.7and 6 8.7. 7 . , 8 . RecursiveFunction: . , GoTopic:EvaluateusingfunctionnotationFindeachvalue.4. 2 .Find 3 . 5. 5 .Find 2 .

8 256. 2 .Find 3 . 7. 3 4 1 .Find 5 and 6 .

, 8. 2 1 6.Find 1 and 2 .

,

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Name: ArithmeticandGeometricSequences 3.5Ready,Set,Go!

ReadyTopic:ArithmeticandgeometricsequencesFindthemissingvaluesforeacharithmeticorgeometricsequence.Thensayifthesequencehasaconstantdifferenceoraconstantratio,andsaywhattheconstantdifference/rateis.1. 5, 10, 15, ___ ___, 25, 30…

Constantdifferenceoraconstantratio?ConstantDifferenceTheconstantdifference/ratiois

2. 20, 10, __ ___, 2.5, ___ . __, …

Constantdifferenceoraconstantratio?ConstantRatioTheconstantdifference/ratiois

3. 2, 5, 8, ___ ___, 14, ___ __, …


4. 30, 24, __ ___, 12, 6, …


SetTopic:RecursiveandexplicitequationsDeterminewhethereachsituationrepresentsanarithmeticorgeometricsequenceandthenfindtherecursiveandexplicitequationforeach.5. 2, 4, 6, 8, …ArithmeticorGeometric?ArithmeticRecursive: , Explicit:

6. 2, 4, 8, 16, …ArithmeticorGeometric?GeometricRecursive: , ⋅ Explicit: or ⋅

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7.Time(days)

NumberofDots

1 32 73 114 15

ArithmeticorGeometric?ArithmeticRecursive: , Explicit:

8.Time(days)

NumberofCells

1 52 83 12.84 20.48

ArithmeticorGeometric?GeometricRecursive: , ⋅ . Explicit: .

9. Scottdecidestoaddrunningtohisexercise

routineandrunsatotalofonemilehisfirstweek.Heplanstodoublethenumberofmilesherunseachweek.

ArithmeticorGeometric?GeometricRecursive: , ⋅ Explicit:

10. Michellelikeschocolatesomuchthatsheeatsiteverydayandshealwayseats3morepiecesthanthepreviousday.Sheate3piecesonday1.


11.Vanessahas$60tospendonridesattheState

Fair.Eachridecost$4.


12. Camiinvested$6,000dollarsintoanaccountthatearns10%interesteachyear.

ArithmeticorGeometric?GeometricRecursive: , ⋅ . Explicit: .

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GoTopic:SolvingsystemsoflinearequationsSolvethesystemofequations.

13.2 104 5

,

14.7 6

3 21 18

InfinitelyManySolutions

15.5 4 36 4 30

,

16.2 3 12

2 4

,

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Name: ArithmeticandGeometricSequences 3.6Ready,Set,Go!ReadyTopic:ConstantRatiosFindtheconstantratioforeachgeometricsequence.1. 2, 4, 8, 16, …

2. , 1, 2, 4, 8, …

3. 5, 10, 20, 40, …

4. 10, 5, 2.5, 1.25, …

SetTopic:RecursiveandexplicitequationsFillintheblanksforeachtableandthenwritetherecursiveandexplicitequationforeachsequence.5. Table1

1 2 3 4 5 5 7 9 11 13

Recursive: , Explicit:

6. Table2 7. Table3 8. Table4

x y x y x y1 2 1 3 1 272 4 2 9 2 93 6 3 27 3 34 4 81 4 1

5 5 243 5

Recursive: Recursive: Recursive:

, , , Explicit: Explicit: Explicit:

or ⋅

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GoTopic:Graphinglinearequationsandlabelingyouraxes.Graphthefollowinglinearequations.Labelyouraxes.9. 4 7

10. 5

11.2 7 10

12. 3 7

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Name: ArithmeticandGeometricSequences 3.7Ready,Set,Go!ReadyTopic:ArithmeticandgeometricsequencesForeachsetofsequences,findthefirstfiveterms.Comparearithmeticsequencesandgeometricsequences.Whichgrowsfaster?When?1. Arithmeticsequence: 1 2,commondifference, 3 Geometricsequence: 1 2,commonratio, 3

Arithmetic:1

2

3

4

5

Geometric:1

2

3

4 54

5

Whichvaluedoyouthinkwillbemore, 100 or 100 ?Why?becausethevalueincreasesmuchfasterwhenmultiplyingtheprevioustermbythesame

valueasopposedtoaddingthesamevaluetothepreviousterm.2. Arithmeticsequence: 1 2,commondifference, 10 Geometricsequence: 1 2,commonratio, 3

Arithmetic:1 2 3 4 5

Geometric:1 2 3 4 5

Whichvaluedoyouthinkwillbemore, 100 or 100 ?Why?becausethevalueincreasesmuchfasterwhenmultiplyingtheprevioustermbythesame

valueasopposedtoaddingthesamevaluetothepreviousterm,evenifthevalueaddedismuchlargerthanthevaluemultipliedbyasseeninthisexample.

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SetTopic:ArithmeticsequencesEachofthetablesbelowrepresentsanarithmeticsequence.Findthemissingtermsinthesequence,showingyourmethod.3. Table1

1 2 3 3 7.5 12

.

4. Table2 5. Table3 6. Table4

1 2 1 24 1 162 10 2 15 2 123 18 3 6 3 84 26 4 4 4 5 0

Topic:GeometricsequencesEachofthetablesbelowrepresentsageometricsequence.Findthemissingtermsinthesequence,showingyourmethod.7. Table1

1 2 3 3 6 12

,

8. Table2 9. Table3 10.Table4

1 2 1 5 1 42 6 2 10 2 123 18 3 20 3 364 54 4 40 4 108 5 324

, , ,

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GoTopic:SequencesDeterminetherecursiveandexplicitequationsforeach(ifthesequenceisnotarithmeticorgeometric,tryyourbest).11.5, 9, 13, 17, … Thissequenceis:Arithmetic,Geometric,Neither

RecursiveEquation: , ExplicitEquation: 12.60, 30, 0, 30, … Thissequenceis:Arithmetic,Geometric,Neither

RecursiveEquation: , ExplicitEquation: 13.60, 30, 15, , … Thissequenceis:Arithmetic,Geometric,Neither

RecursiveEquation: , ExplicitEquation:

14.

(Thepercentageoftilesshadedblack)

Thissequenceis:Arithmetic,Geometric,Neither

RecursiveEquation: , ExplicitEquation:

15.4, 7, 12, 19, … Thissequenceis:Arithmetic,Geometric,Neither

RecursiveEquation: , ExplicitEquation: **Note:Studentsarenotexpectedtobeabletowritetherecursiveorexplicitequationsforquestion15atthispoint**

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Name: ArithmeticandGeometricSequences 3.8Ready,Set,Go!ReadyTopic:Comparinglinearequationsandarithmeticsequences1. Describesimilaritiesanddifferencesbetweenlinearequationsandarithmeticsequences.

Similarities Differences Bothhaveaconsistentchangefor

everyinterval. Bothcanberepresentedasfunctions

ofavariable. Bothhavepointslieonaline.

Linearequationsrepresentallsolutionstoallx‐values,whereasarithmeticsequenceschooseonlyspecificvalues.

SetTopic:representationsofarithmeticsequencesUsethegiveninformationtocompletetheotherrepresentationsforeacharithmeticsequence.2. RecursiveEquation:

,

ExplicitEquation:

Table:Days Cost1 82 163 244 32

CreateaContext:Itcosts$8perdaytorentakayak.

Graph:

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3. RecursiveEquation:1 4, 1 3

ExplicitEquation:

Table:Hour Cost1 42 73 104 13

CreateaContext:Itcostsaflatfeeof$1tocheckoutskates,andthen$3perhourfortherental.

Graph:

4. RecursiveEquation:

,

ExplicitEquation:

4 5 1

Table:Days Cost1 42 93 144 19

CreateaContext:ItCosts$4torentsnorkelgearonthefirstday,andthen$5everydayafterthat.

Graph:

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5. RecursiveEquation:,

ExplicitEquation:

Table:Row #ofseats1 142 163 184 20

CreateaContext:Janetwantstoknowhowmanyseatsareineachrowofthetheater.Jamalletsherknowthateachrowhas2seatsmorethantherowinfrontofit.Thefirstrowhas14seats.

Graph:

GoTopic:WritingexplicitequationsGiventherecursiveequationforeacharithmeticsequence,writetheexplicitequation.6. 1 2; 1 8

7. 5 1 ; 1 0

8. 1 1; 1

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Name: ArithmeticandGeometricSequences 3.9Usethegiveninformationtostateasmuchaspossibleabouteachsequence.Youranswershouldinclude:typeofsequence,thecommondifferenceorcommonration,atableofatleast5terms,agraph,therecursiverule,andtheexplicitrule.1. Type:Arithmetic 1 2

2 53 84 115 14

Commondifference/ratio: 3

Recursiverule:1 2, 1 3

Explicitrule:

2. Type:Geometric 0 3

1 62 123 244 48


Recursiverule:,

Explicitrule:3 ∙ 2

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3. Type:Arithmetic 1 3

2 53 74 95 11


Recursiverule:,

Explicitrule:

4. Type:Geometric 1 40

2 203 104 55 2.5

CommonRatio=

Recursiverule:,

Explicitrule:

5. Explainhowyoutellifasequenceisarithmeticandifasequenceisgeometric.

Ifasequenceisarithmeticthereisacommondifferencethatisaddedorsubtractedfromthepreviousvaluetogetthenextvalue,whereas,ifasequenceisgeometricthereisacommonratiothatismultipliedbythepreviousvaluetogetthenextvalue.Inaddition,anarithmeticsequencecreatesalinearfunction,whereasageometricsequencecreatesanexponentialfunction.

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For#6‐8,determineifeachsequenceisarithmeticorgeometric.Findthevaluesofthenexttwoterms.Thenwritetheexplicitandrecursiveformulasforeachsequence.6. 90, 30, 10, , …

ArithmeticorGeometricNext2terms: ,

RecursiveFormula: , ExplicitFormula:

7. 42, 34, 26, 18, …

ArithmeticorGeometricNext2terms:10,2RecursiveFormula: , ExplicitFormula:

8. 6, 13, 20, 27, …

ArithmeticorGeometricNext2terms:34,41RecursiveFormula: , ExplicitFormula:

9. Findthemissingtermsofthearithmeticsequencebelow.Besuretoshowallwork.

n 1 2 3 4 5 6 7 18