sequences and series 3.1h set, go!msshultis.weebly.com/uploads/1/0/9/3/10930910/2016-17...te‐18...
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SDUHSDMath1Honors
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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SDUHSDMath1Honors
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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SDUHSDMath1Honors
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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SDUHSDMath1Honors
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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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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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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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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SDUHSDMath1Honors
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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SDUHSDMath1Honors
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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SDUHSDMath1Honors
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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SDUHSDMath1Honors
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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SDUHSDMath1Honors
GoTopic:WritingexplicitequationsGiventherecursiveequationforeacharithmeticsequence,writetheexplicitequation.9. 1 2; 1 8
10. 5 1 ; 1 0
11. 1 1; 1
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SDUHSDMath1Honors
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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SDUHSDMath1Honors
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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SDUHSDMath1Honors
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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SDUHSDMath1Honors
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.