Sequences and Series 3.1H Set, Go!msshultis.weebly.com/uploads/1/0/9/3/10930910/2016-17...TE‐18...

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Name: SequencesandSeries 3.1HReady,Set,Go!ReadyTopic:Findingvaluesforapattern1. BobCooperwasbornin1900.By1930hehad3sons,allwiththeCooperlastname.By1960eachof

Bob’s3boyshadexactly3sonsoftheirown.Bytheendofeach30yeartimeperiod,thepatternofeachCooperboyhavingexactly3sonsoftheirowncontinued.HowmanyCoopersonswereborninthe30yearperiodbetween1960and1990?27

2. Createadiagramthatwouldshowthispattern.

Year 1900 1930 1960 1990 2020#ofsons 1 3 9 27 81

3. PredicthowmanyCoopersonswillbebornbetween1990and2020,ifthepatterncontinues. 814. TrytowriteanequationthatwouldhelpyoupredictthenumberofCoopersonsthatwouldbeborn

between2020and2050.Ifyoucan’tfindtheequation,explainitinwords.

5. HowmanyCoopersonswereborninallfrom1900to2020? 121Topic:FunctionNotationForeachofthefollowing,find , and 6. 2 7. 3 2 , , , , 8. 2 1 3

, , Completeeachtable.9.

Term 1st 2nd 3rd 4th 5th 6th 7th 8thValue 66 50 34 18

10.

Term 1st 2nd 3rd 4th 5th 6th 7th 8thValue 3 9 27 81

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SetTopic:CompletingatableFillinthetable.Thenwriteasentenceexplaininghowyoufiguredoutthevaluestoputineachcell.Explainhowtofigureoutwhatwillbeincell#8.11.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.

12.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

Topic:EvaluatingequationsEvaluatethefollowingequationswhen , , , , .Organizeyourinputsandoutputsintoatableofvaluesforeachequation.Letxbetheinputandybetheoutput.13. 4 14. 3 15. 3

x y1 42 163 44 2565 1024

x y12345

x y1 2 3 4 5

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SDUHSDMath1Honors

GoTopic:GoodviewingwindowWhensketchingagraphofafunction,itisimportantthatweseekeypoints.Forlinearfunctions,wewantawindowthatshowsimportantinformationrelatedtothestory.Often,thismeansincludingboththex‐andy‐interceptsFindanappropriategraphingwindowforeachofthefollowinglinearfunctions.Fillintheblanksshowingtheloweranduppervaluesandincludethescaleforeachaxis.YoumayuseanonlinegraphingutilitysuchasDesmos(https://www.desmos.com/calculator)orMATHPAPA(https://www.mathpapa.com/calc.html?q=)Answersmayvary.Sampleanswersprovidedbelow:16. 1 x: , byy: , x‐scale: y‐scale:

17. 7 3 14 x: , byy: , x‐scale:1 y‐scale:1

18. 3 5 12 x: , byy: , x‐scale:1 y‐scale:1

19. 15 10 45 x: , byy: , x‐scale:2 y‐scale:25

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Name: SequencesandSeries 3.2HReady,Set,Go!ReadyTopic:Writetheequationofalinegiventwopoints.Writeanequationofthelinethatgoesthroughthegiventwopoints.1. 5, 2 and 7, 0 2. 2, 4 and 2, 6

SetTopic:RecursiveandexplicitfunctionsofarithmeticsequencesBelowyouaregivenvarioustypesofinformation.Writetherecursiveandexplicitfunctionsforeachsequence.Finally,grapheachsequence,makingsureyouclearlylabelyouraxes.3. 2, 4, 8, 16, …

Recursive:

, Explicit:

4. EachdayTaniadecidestodosomethingnicefor2strangers.WriterecursiveandexplicitequationsthatrepresentthenumberofstrangersTaniathatdoessomethingniceforeachday(nottotalnumberofstrangers).

Recursive:

, Explicit:

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5. Clairehas$300inanaccount.Shedecidessheisgoingtotakeouthalfofwhat’sleftinthereattheendofeachmonth.

Recursive:

, Explicit:

6. Taniacreatesachainletterandsendsittofourfriends.Eachdayeachfriendistheninstructedtosendittofourfriendsandsoforth.

Recursive:

, Explicit:

7.

Recursive:

, Explicit:

Topic:Summationnotationforaseries8. Writeoutwhatismeantby:

a. ∑ 3 b. ∑

9. Writethefollowinginsummationnotation: a. 3 3 3 3 b. 2 4 6 8 10 12

∑ ∑ 10.Arethefollowingseriesequivalent?Explainyourreasoning. ∑ and ∑ Yes.Bothsequencesrepresent: ⋯

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SDUHSDMath1Honors

GoTopic:ArithmeticandgeometricsequencesDetermineifthefollowingsequencesarearithmetic,geometric,both,orneither.11.109,94,79,64 Arithmetic12.Christinedid41sit‐upsonTuesday,44sit‐upsonWednesday,46sit‐upsonThursday,47sit‐upson

Friday. Neither13.1,9,81,729,… Geometric14.Whilesortingchangeintoapiggybank,Ruthput14coinsinthefirstpiggybank,14coinsinthesecond

piggybank,14coinsinthethirdpiggybank,and14coinsinthefourthpiggybank. Both15.6, 24, 144, 864 Geometric16.Abookshelfhas7shelvesofdifferentwidths.Eachshelfisnarrowerthantheshelfbelowit.Thebottom

threeshelvesare36in.,31in.,and26in.wide.Theshelfwidthsdecreasebythesameamountfrombottomtotop.

a. Whatisthewidthofthetopshelf? 6inchesb. Whatisthetotalshelfspaceofallsevenshelves? 147inches

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Name: SequencesandSeries 3.3HReady,Set,Go!ReadyTopic:ArithmeticandgeometricsequencesFindthemissingvaluesforeacharithmeticorgeometricsequence.Thensayifthesequencehasaconstantdifferenceoraconstantratio,andsaywhattheconstantdifference/rateis.1. 5,10,15,______,25,30… Constantdifferenceoraconstantratio? ConstantDifference Theconstantdifference/ratiois

2. 20,10,______,2.5,______,… Constantdifferenceoraconstantratio? ConstantRatio Theconstantdifference/ratiois

3. 2,5,8,______,14,_____,… Constantdifferenceoraconstantratio? ConstantDifference Theconstantdifference/ratiois

4. 30,24,_____,12,6,… Constantdifferenceoraconstantratio? ConstantDifference Theconstantdifference/ratiois

SetTopic:DeterminerecursiveequationsTwoconsecutivetermsinanarithmeticsequencearegiven.Findtheconstantdifferenceandtherecursiveequation.5. If 3 5and 4 8.

5 , 6 RecursiveFunction: , 6. If 2 20and 3 12. 4 , 5 RecursiveFunction: ,

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SDUHSDMath1Honors

Topic:RecursiveandexplicitequationsDeterminewhethereachsituationrepresentsanarithmeticorgeometricsequenceandthenfindtherecursiveandexplicitequationforeach.7. 2, 4, 6, 8, … ArithmeticorGeometric?Arithmetic Recursive: , Explicit:

8.Time(days)

NumberofCells

1 52 83 12.84 20.48

ArithmeticorGeometric?Geometric Recursive: . , Explicit: .

9. Camiinvested$6,000dollarsintoanaccountthatearns10%interesteachyear.

ArithmeticorGeometric?Geometric Recursive: , . Explicit: .

10. Scottdecidestoaddrunningtohisexerciseroutineandrunsatotalofonemilehisfirstweek.Heplanstodoublethenumberofmilesherunseachweek.

ArithmeticorGeometric?Geometric Recursive: , Explicit:

11.Vanessahas$60tospendonridesattheStateFair.Eachridecost$4.

ArithmeticorGeometric?Arithmetic Recursive: , Explicit:

12. Michellelikeschocolatesomuchthatsheeatsiteverydayanditalways3morepiecesthanthepreviousday.Sheate3piecesonday1.

ArithmeticorGeometric?Arithmetic Recursive: , Explicit:

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SDUHSDMath1Honors

GoTopic:EvaluateusingfunctionnotationFindeachvalue.13. 5 .Find 2 .

2514. 2 .Find 3 .

15. 3 4 1 .Find 5 and 6 .

, Topic:SolvingsystemsoflinearequationsSolvethesystemofequationsusingamatrix.

16.2 104 5

,

17.7 6

3 21 18

InfinitelyManySolutions

18.5 4 36 4 30

,

19.2 3 12

2 4

,

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SDUHSDMath1Honors

Name: SequencesandSeries 3.4HReady,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

n 1 2 3 4 5 5 7 9 11 13

Recursive: , Explicit:

6. Table2 7. Table3 8. Table4

n n n1 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 or ⋅

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SDUHSDMath1Honors

Topic:SubscriptnotationforsequencesOthertextbooksmayusesubscriptnotationtowriterulesforsequences.Usetheexamplesbelowtowritetherecursiveandexplicitrulesforthefollowingsequences.ExampleSequence FunctionNotation SubscriptNotation

3, 5, 7, 9, …Recursive: 1 3, 1 2Explicit: 2 1

Recursive: 3, 2Explicit: 2 1

3, 9, 27, 81, …Recursive: 1 3, 1 ⋅ 3Explicit: 3

Recursive: 3, ⋅ 3Explicit: 3

9. 22, 19, 16, … 10.1, 5, 25, … Recursive: , Recursive: , ⋅ Explicit: Explicit: Topic:Arithmeticseries11.Findthesumofthefirst12termsofthesequence 2 10 12.Findthesum:∑ 3 1 67213.Findthesumofthefirst150termsofthesequence20,15,10,5,… , 14.Findthesumofthefirst200evennumbers. 402,00015.TheAgnesiHighSchoolauditoriumhasexactly26rowsofseats.Therowsarelabeled,inorder,fromthe

frontoftheauditoriumtothebackfromAthroughZ.Thereare8seatsintherowA.Eachrowafterthefirstrowhastwomoreseatsthanthepreviousrow.Thereare10seatsinrowB,12seatsinrowCandsoon.a. HowmanyseatsarethereinrowZ?

Thereare58seatsinRowZ.

b. WhatisthetotalnumberofseatsintheAgnesiHighSchoolauditorium?

858seats

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16.Thefirstfigurecontainsonesegment.Foreachsuccessivefigure,sixsegmentsreplaceeachsegment.Thisisanexampleofafractal.a. Howmanysegmentsareineachofthefirstfourfiguresofthesequence? , , , b. Writearecursivedefinitionforthesequence. , ⋅

GoTopic:Graphinglinearequationsandlabelingyouraxes.Graphthefollowinglinearequations.Labelyouraxes.17. 4 7

18. 5

19.2 7 10

20. 3 7

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SDUHSDMath1Honors

Name: SequencesandSeries 3.5HReady,Set,Go!ReadyTopic:ArithmeticandgeometricsequencesForeachsetofsequences,findthefirstfiveterms.Comparearithmeticsequencesandgeometricsequences.Whichgrowsfaster?When?1. Arithmeticsequence: 1 2,commondifference,d =3 Geometricsequence: 1 2,commonratio,r=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,d =10 Geometricsequence: 1 2,commonratio,r=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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SDUHSDMath1Honors

SetTopic:ArithmeticsequencesEachofthetablesbelowrepresentsanarithmeticsequence.Findthemissingtermsinthesequence,showingyourmethod.3. Table1

n 1 2 3 3 7.5 12

.

4. Table2 5. Table3 6. Table4

n n n1 2 1 24 1 162 10 2 15 2 123 18 3 6 3 84 26 4 4 4 5 0

7. Table5

n 2 3 4 5 6 32 27 22 17 12

Topic:GeometricsequencesEachofthetablesbelowrepresentsageometricsequence.Findthemissingtermsinthesequence,showingyourmethod.8. Table1

n 1 2 3 3 6 12

,

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9. Table2 10.Table3 11.Table4

n n n1 2 1 5 1 42 6 2 10 2 123 18 3 20 3 364 54 4 40 4 108 5 324

, , ,

12.Table5

n 3 4 5 6 18 54 162 486

,

GoTopic:SequencesDeterminetherecursiveandexplicitequationsforeach(ifthesequenceisnotarithmeticorgeometric,tryyourbest).Expressanswersinbothfunctionsubscriptnotation.13.5, 9, 13, 17, … Thissequenceis:Arithmetic,Geometric,Neither

RecursiveEquation: , ExplicitEquation: ,

14.60, 30, 0, 30, … Thissequenceis:Arithmetic,Geometric,Neither

RecursiveEquation: , ExplicitEquation: ,

15.60, 30, 15, , … Thissequenceis:Arithmetic,Geometric,Neither

RecursiveEquation: , ExplicitEquation:

, ⋅

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16.

(Thepercentageoftilesshadedblack)

Thissequenceis:Arithmetic,Geometric,Neither

RecursiveEquation: , ExplicitEquation:

, ⋅

17.4, 7, 12, 19, … Thissequenceis:Arithmetic,Geometric,Neither

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

18. Writethefollowingseriesinsummationnotation:20 14 8 2 4 10 ∑ 19. Findthesumof∑ 2 60

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Name: SequencesandSeries 3.6HReady,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:

Topic:ApplicationofarithmeticandgeometricseriesWriteaseriesrepresentationusingsummationnotationforeachscenarioandthenfindthesum.6. Logsarestackedinapilewith24logsonthebottomrowand15onthetoprow.Thereare10rowsinall

witheachrowhavingonemorelogthantheoneaboveit.Howmanylogsareinthestack? ∑ 7. Eachhour,agrandfatherclockchimesthenumberoftimesthatcorrespondstothetimeofday.For

example,at3:00,itwillchime3times.Howmanytimesdoestheclockchimeinaday? ⋅ ∑ 8. Acompanyisofferingajobwithasalaryof$30,000forthefirstyearanda5%raiseeachyearafterthat.

Ifthat5%raisecontinueseveryyear,findthetotalamountofmoneyyouwillhaveearnedbytheendofyour5thyear.

∑ , . $ , .

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GoTopic:WritingexplicitequationsGiventherecursiveequationforeacharithmeticsequence,writetheexplicitequation.9. 1 2; 1 8

10. 5 1 ; 1 0

11. 1 1; 1

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Name: SequencesandSeries ReviewUsethegiveninformationtostateasmuchaspossibleabouteachsequence.Youranswershouldinclude:typeofsequence,thecommondifferenceorcommonration,atableofatleast5terms,agraph,therecursiverule,andtheexplicitrule.1. Type:Arithmetic

n1 22 53 84 115 14

Commondifference/ratio: 3

Recursiverule:1 2, 1 3

Explicitrule:

2. Type:Geometric

n0 31 62 123 244 48

Commondifference/ratio:2

Recursiverule:,

Explicitrule:3 ∙ 2

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3. Type:Arithmetic

n1 32 53 74 95 11

Commondifference/ratio: 2

Recursiverule:,

Explicitrule:

4. Type:Geometric

n1 402 203 104 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

10.Findthemissingtermsofthegeometricsequencebelow.Besuretoshowallwork.

n 1 2 3 4 5 6 . 7 56

,

11.Findthemissingtermsofthegeometricsequencebelow.Besuretoshowallwork.

n 1 2 3 4 5 6 972 12

,

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12.Findthesumofthefirst50multiplesof6.

,

13. Achildbuildingatowerwithblocksuses15forthebottomrow.Eachrowhas2fewerblocksthanthe

previousrow.Supposethatthereare8rowsinthetower.

a. Howmanyblocksareusedforthetoprow? 1block

b. Whatisthetotalnumberofblocksinthetower? 64blocks14.Whatisthecommonratiooftheseriesmodeledby∑ 4 3 ? 15.Howmanytermsareinthegeometricsequencehavingafirsttermof2,alasttermof32,andacommon

ratioof 2? 5terms16.Asnailiscrawlingstraightupawall.Thefirsthouritclimbs16inches,thesecondhouritclimbs12

inches,andeachsucceedinghour,itclimbsonlythree‐fourthsthedistanceitclimbedtheprevioushour.Assumethepatterncontinues.

a. Howfardoesthesnailclimbduringthefifthhour?

. inches

b. Whatisthetotaldistancethesnailhasclimbedinfivehours?

. inches

c. Expressthetotaldistancewithsummationnotation.

∑

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Module4HIntroductionHomeworkThefollowingproblemsareintendedforstudentstoworkonaftertheModule3HTest.ThefirstfourproblemsintroducethebeginningtasksofModule4H.Theproblemsaremeanttobedoneontheirownandwillbediscussedduringthewarmupthenextclass.Thefollowingpageisblankfortheteachertocopyandgivetoeachstudentafterthetest.Belowarethesolutionsforthechallengeproblems.For#1‐4,createamathematicalmodelforthatincludesatable,graph,andequation:1. Mylittlesister,Savannah,isthreeyearsold.Shehasapiggybankthatshewantstofill.Shestarted

withfivepenniesandeachdaywhenIcomehomefromschool,sheisexcitedwhenIgiveherthreepenniesthatareleftoverfrommylunchmoney.Createamathematicalmodelforthenumberofpenniesinthepiggybankondayn.

whereprepresentsthenumberofpenniesSavannahhasandnrepresentsthenumber

ofdaysthathavepassed.

Checkfortable&graph.2. Ourfamilyhasasmallpoolforrelaxinginthesummerthatholds1500gallonsofwater.Idecidedtofill

thepoolforthesummer.WhenIhad5gallonsofwaterinthepool,IdecidedthatIdidn’twanttostandoutsideandwatchthepoolfill,soIhadtofigureouthowlongitwouldtakesothatIcouldleave,butcomebacktoturnoffthewaterattherighttime.Icheckedtheflowonthehoseandfoundthatitwasfillingthepoolatarateof2gallonseveryfiveminutes.Createamathematicalmodelforthenumberofgallonsofwaterinthepoolattminutes.

. Checkfortable&graph.

3. I’mmoresophisticatedthanmylittlesistersoIsavemymoneyinabankaccountthatpaysme3%

interestonthemoneyintheaccountattheendofeachmonth.(IfItakemymoneyoutbeforetheendofthemonth,Idon’tearnanyinterestforthemonth.)Istartedtheaccountwith$50thatIgotformybirthday.CreateamathematicalmodeloftheamountofmoneyIwillhaveintheaccountaftermmonths.

.

Checkfortable&graph.

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4. Attheendofthesummer,Idecidetodraintheswimmingpoolthatholds1500gallonsofwater.Inoticedthatitdrainsfasterwhenthereismorewaterinthepool.Thatwasinterestingtome,soIdecidedtomeasuretherateatwhichitdrains.Ifoundthatitwasdrainingatarateof3%everyminute.Createamathematicalmodelofthegallonsofwaterinthepoolattminutes.

. Checkfortable&graph.

Module3ChallengeProblems5. Considerthepatternofsquaregridsshown.ThesumofthenumbersinthesquaregridatStage3is27.

Ifthepatterncontinues,whatwillbethesumofthenumbersinthesquaregridatStage7?

ThesumofthenumbersattheStages1through4are1,8,27and64,respectively.Noticethatthesenumbersareallperfectcubes: , , , .Thesesumsformasequencewherethenthtermofthesequenceis .Therefore,thesumofthenumbersatStage7willbe .

6. Thesumofalistofsevenpositiveintegersis42.Themean,medianandmodeareconsecutiveintegers,

insomeorder.Whatisthelargestpossibleintegerinthelist?

Wearetoldthatthesumofthesevenintegersis42,soitfollowsthatthesenumbershaveameanof .Thus,therearethreepossibilitiesfortheconsecutivemeasuresofcentraltendency4,5and6;5,6and7;6,7and8.Butthevalues4,5and6willyieldthelargestpossibleintegerinsuchalist.Sincethemeanis6,themodeandmediancouldbe4and5,respectively,orviceversa.Suppose,4isthemodeand5isthemedian,togetthelargestintegerwewouldhavetheseries

.Thisgivesavalueof .Suppose4isthemedianand5isthemode.Sincethemeandoesnotneedtobeoneoftheintegersinthelist,togetthelargestintegerwewouldhavetheseries .Thisgivesavalueof .Therefore,thelargestpossibleintegerinthelistis22.