Algebra 2 Unit 5: Sequences & Series - Ms. Talhami - Home€¦ · Algebra 2 Unit 5: Sequences & Series Name_____ Algebra 2 Unit 5: Sequences & Series Ms. Talhami 2 SEQUENCES COMMON

Algebra2Unit5:Sequences&Series

Ms.Talhami 1


Name_________________


Ms.Talhami 2

SEQUENCES COMMONCOREALGEBRAII

InCommonCoreAlgebraI,youstudiedsequences,whichareorderedlistsofnumbers.Sequencesareextremelyimportantinmathematics,boththeoreticalandapplied.Asequenceisformallydefinedasafunctionthathasasitsdomainthesetthesetofpositiveintegers,i.e.{ }1, 2, 3, ..., n .

Exercise#1:Asequenceisdefinedbytheequation ( ) 2 1a n n= − .(c) Explainwhytherewillnotbeatermthathasavalueof70.Recall that sequences can also be described by using recursive definitions. When a sequence is definedrecursively,termsarefoundbyoperationsonpreviousterms.

Exercise#2:Asequenceisdefinedbytherecursiveformula: ( ) ( )1 5f n f n= − + with ( )1 2f = − .Exercise#3:Determinearecursivedefinition,intermsof ( )f n ,forthesequenceshownbelow.Besuretoincludeastartingvalue.

5,10,20,40,80,160,…Exercise#4:Fortherecursivelydefinedsequence ( )21 2n nt t −= + and 1 2t = ,thevalueof 4t is (1)18 (3)456 (2)38 (4)1446

(a) Generate the first five termsof this sequence.Labeleachtermwithproperfunctionnotation.

(b)Determine the value of . Hint – thinkabouthowmanytimesyouhaveadded5to.

(a) Find the first three terms of this sequence,denotedby .

(b)Whichtermhasavalueof53?


Ms.Talhami 3

Exercise#5: Oneofthemostwell-knownsequencesistheFibonacci,whichisdefinedrecursivelyusingtwopreviousterms.Itsdefinitionisgivenbelow.

( ) ( ) ( )1 2f n f n f n= − + − and ( ) ( )1 1 and 2 1f f= =

Generatevaluesfor ( ) ( ) ( ) ( )3 , 4 , 5 , and 6f f f f (inotherwords,thennextfourtermsofthissequence).Itisoftenpossibletofindalgebraicformulasforsimplesequence,andthisskillshouldbepracticed.Exercise#6:Findanalgebraicformula ( )a n ,similartothatinExercise#1,foreachofthefollowingsequences.

Recallthatthedomainthatyoumapfromwillbetheset{ }1, 2, 3, ..., n .

(a) 4, 5, 6, 7, ... (b) 2, 4, 8,16, ... (c) 5 5 55, , , , ...2 3 4

(d) 1, 1, 1, 1, ...− − (e)10,15,20,25,… (f) 1 1 11, , , , ...4 9 16

Exercise#7:Whichofthefollowingwouldrepresentthegraphofthesequence 2 1na n= + ?Explainyourchoice.(1) (2) (3) (4)

Explanation:

y

n

y

n

y

n

y

n


Ms.Talhami 4

SEQUENCES COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Giveneachofthefollowingsequencesdefinedbyformulas,determineand label thefirst fourterms. A

varietyofdifferentnotationsisusedbelowforpracticepurposes.

(a) ( ) 7 2f n n= + (b) 2 5na n= − (c) ( ) ( )23

nt n = (d) 1

1nt n=

+

2. Sequencesbelowaredefinedrecursively.Determineandlabelthenextthreetermsofthesequence.

(a) ( ) ( ) ( )1 4 and 1 8f f n f n= = − + (b) ( ) ( ) ( )11 and 1 242a n a n a= − ⋅ =

(c) 1 2n nb b n−= + with 1 5b = (d) ( ) ( ) 22 1f n f n n= − − and ( )1 4f = 3. Giventhesequence7,11,15,19,...,whichofthefollowingrepresentsaformulathatwillgenerateit?

(1) ( ) 4 7a n n= + (3) ( ) 3 7a n n= +

(2) ( ) 3 4a n n= + (4) ( ) 4 3a n n= +

4. Arecursivesequenceisdefinedby 1 1 1 22 with 0 and 1n n na a a a a+ −= − = = .Whichofthefollowingrepresentsthevalueof 5a ?

(1)8 (3)3

(2) 7− (4)45. Whichofthefollowingformulaswouldrepresentthesequence10,20,40,80,160,…

(1) 10nna = (3) ( )5 2 nna =

(2) ( )10 2 nna = (4) 2 10na n= +


Ms.Talhami 5

6. Foreachofthefollowingsequences,determineanalgebraicformula,similartoExercise#4,thatdefinesthesequence.

(a)5,10,15,20,… (b)3,9,27,81,… (c) 1 2 3 4, , , , ...2 3 4 5

7. Foreachofthefollowingsequences,statearecursivedefinition.Besuretoincludeastartingvalueorvalues.

(a)8,6,4,2,… (b)2,6,18,54,… (c) 2, 2, 2, 2, ...− − APPLICATIONS

8. Atilingpatterniscreatedfromasinglesquareandthenexpandedasshown.Ifthenumberofsquaresineachpatterndefinesasequence,thendeterminethenumberofsquaresintheseventhpattern.Explainhowyouarrivedatyourchoice.Canyouwritearecursivedefinitionforthepattern?

REASONING

9. ConsiderasequencedefinedsimilarlytotheFibonacci,butwithaslighttwist:

( ) ( ) ( )1 2f n f n f n= − − − with ( ) ( )1 2 and 2 5f f= =

Generateterms ( ) ( ) ( ) ( ) ( ) ( )3 , 4 , 5 , 6 , 7 , and 8f f f f f f .Then,determinethevalueof ( )25f .


Ms.Talhami 6

ARITHMETIC AND GEOMETRIC SEQUENCES COMMONCOREALGEBRAII

In CommonCoreAlgebra I, you studied two particular sequences known asarithmetic (based on constantadditiontogetthenextterm)andgeometric(basedonconstantmultiplyingtogetthenextterm).Inthislesson,wewillreviewthebasicsofthesetwosequences.Exercise#1:Generatethenextthreetermsofthegivenarithmeticsequences.

(a) ( ) ( )1 6f n f n= − + with ( )1 2f = (b) 1 11 3 and 2 2n na a a−= + =

Exercise#2:Forsomenumbert,thefirstthreetermsofanarithmeticsequenceare2t ,5 1t − ,and6 2t + .Whatisthenumericalvalueofthefourthterm?Hint:firstsetupanequationthatwillsolvefort.Itisimportanttobeabletodetermineageneraltermofanarithmeticsequencebasedonthevalueoftheindexvariable(thesubscript).Thenextexercisewalksyouthroughthethinkingprocessinvolved.Exercise#3:Consider 1 13 with 5n na a a−= + = .

ARITHMETICSEQUENCERECURSIVEDEFINITION

Given ,then orgiven

wherediscalledthecommondifferenceandcanbepositiveornegative.

(a) Determinethevalueof . (b)Howmanytimeswas3addedto5 inordertoproduce ?

(c) Useyourresultfrompart(b)toquicklyfindthevalueof .

(d)Writeaformulaforthe termofanarithmeticsequence, ,basedonthefirstterm, ,dandn.


Ms.Talhami 7

Exercise#4:Giventhat 1 46 and 18a a= = aremembersofanarithmeticsequence,determinethevalueof 20a.Geometricsequencesaredefinedverysimilarlytoarithmetic,butwithamultiplicativeconstantinsteadofanadditiveone.Exercise#5:Generatethenextthreetermsofthegeometricsequencesgivenbelow.(a) 1 4 and 2a r= = (b) ( ) ( ) 11 3f n f n= − ⋅ with ( )1 9f = (c) 1 12 with 3 2n nt t t−= ⋅ =

And,likearithmetic,wealsoneedtobeabletodetermineanygiventermofageometricsequencebasedonthefirstvalue,thecommonratio,andtheindex.Exercise#6:Consider 1 12 and 3n na a a −= = ⋅ .

GEOMETRICSEQUENCERECURSIVEDEFINITION

Given orgiven ,then

whereriscalledthecommonratioandcanbepositiveornegativeandisoftenfractional.

(a) Generatethevalueof . (b)Howmanytimesdidyouneedtomultiply2by3inordertofind .

(c) Determinethevalueof . (d)Writeaformulaforthe termofageometricsequence, ,basedonthefirstterm, ,randn.


Ms.Talhami 8

ARITHMETIC AND GEOMETRIC SEQUENCES COMMONCOREALGEBRAIIHOMEWORK

FLUENCY1. Generatethenextthreetermsofeacharithmeticsequenceshownbelow.

(a) 1 2 and 4a d= − = (b) ( ) ( ) ( )1 8 with 1 10f n f n f= − − = (c) 1 23, 1a a= = 2. Inanarithmeticsequence 1 7n nt t −= + .If 1 5t = − determinethevaluesof 4 20 and t t .Showthecalculations

thatleadtoyouranswers.3. If 4, 2 5, and 4 3x x x+ + + representthefirstthreetermsofanarithmeticsequence,thenfindthevalueof

x.Whatisthefourthterm?4. If ( )1 12f = and ( ) ( )1 4f n f n= − − thenwhichofthefollowingrepresentsthevalueof ( )40f ?

(1) 148− (3) 144− (2) 140− (4) 172− 5. Inanarithmeticsequenceofnumbers 1 64 and 46a a= − = .Whichofthefollowingisthevalueof 12a ?

(1)120 (3)92 (2)146 (4)1066. Thefirsttermofanarithmeticsequencewhosecommondifferenceis7andwhose22ndtermisgivenby

22 143a = iswhichofthefollowing?

(1) 25− (3)7 (2) 4− (4) 28


Ms.Talhami 9

7. Generatethenextthreetermsofeachgeometricsequencedefinedbelow.

(a) 1 8 with 1a r= − = − (b) 132n na a −= ⋅ and 1 16a = (c) ( ) ( ) ( )1 2 and 1 5f n f n f= − ⋅− =

8. Given that 1 25 and 15a a= = are the first two termsof a geometric sequence, determine the valuesof

3 10 and a a .Showthecalculationsthatleadtoyouranswers.9. Inageometricsequence,itisknownthat 1 1a = − and 4 64a = .Thevalueof 10a is (1) 65,536− (3)512 (2) 262,144 (4) 4096− APPLICATIONS10.TheKochSnowflakeisamathematicalshapeknownasafractalthathasmanyfascinatingproperties.Itis

createdbyrepeatedlyformingequilateraltrianglesoffofthesidesofotherequilateraltriangles.Itsfirstsixiterationsareshowntotheright.Theperimetersofeachofthefiguresformageometricsequence.

(a) If the perimeter of the first snowflake (the equilateral

triangle) is 3, what is the perimeter of the second snowflake?Note:thedashedlinesinthesecondsnowflakearenot to be counted towards the perimeter. They are onlytheretoshowhowthesnowflakewasconstructed.

(b)Giventhattheperimetersformageometricsequence,what

is the perimeter of the sixth snowflake? Express youranswertothenearesttenth.

(c) Ifthethisprocesswasallowedtocontinueforever,explainwhytheperimeterwouldbecomeinfinitely

large.


Ms.Talhami 10

SUMMATION NOTATION COMMONCOREALGEBRAII

Muchofourworkinthisunitwillconcernaddingthetermsofasequence.Inordertospecifythisadditionorsummarizeit,weintroduceanewnotation,knownassummationorsigmanotationthatwillrepresentthesesums.Thisnotationwillalsobeusedlaterinthecoursewhenwewanttowriteformulasusedinstatistics.Exercise#1:Evaluateeachofthefollowingsums.

(a) 5

32

ii

=∑ (b)

32

1kk

=−∑ (c)

2

2

2 jj=−∑

(d) ( )5

11 i

i=−∑ (e) ( )

2

02 1

kk

=

+∑ (f) ( )3

11

ii i

=

+∑

Exercise#2:Whichofrepresentsthevalueof4

1

1i i=∑ ?

(1) 110 (3) 2512

(2) 9 4 (4) 3124

SUMMATION(SIGMA)NOTATION

where i iscalledthe indexvariable,whichstartsatavalueofa,endsatavalueofn,andmovesbyunitincrements(increaseby1eachtime).


Ms.Talhami 11

Exercise#3:Considerthesequencedefinedrecursivelyby 1 2 1 22 and 0 and 1n n na a a a a− −= + = = .Findthe

valueof7

4i

ia

=∑

Itisalsogoodtobeabletoplacesumsintosigmanotation.Theseanswers,though,willnotbeunique.Exercise#4:Expresseachsumusingsigmanotation.Useiasyourindexvariable.First,consideranypatternsyounoticeamongstthetermsinvolvedinthesum.Then,worktoputthesepatternsintoaformulaandsum.

(a)9 16 25 100+ + + ⋅ ⋅ ⋅+ (b) 6 3 0 3 15− + − + + + ⋅ ⋅ ⋅ (c) 1 1 1 5 62525 5+ + + + ⋅ ⋅ ⋅

Exercise#5:Whichofthefollowingrepresentsthesum3 6 12 24 48+ + + + ?

(1)5

13i

i=∑ (3)

41

06i

i

−

=∑

(2) ( )4

03 2 i

i=∑ (4)

48

3ii

=∑

Exercise#6:Somesumsaremoreinterestingthanothers.Determinethevalueof99

1

1 11i i i=

⎛ ⎞−⎜ ⎟+⎝ ⎠∑ .Showyour

reasoning.Thisisknownasatelescopingseries(orsum).


Ms.Talhami 12

SUMMATION NOTATION COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Evaluateeachofthefollowing.Placeanynon-integeranswerinsimplestrationalform.

(a)5

24

ii

=∑ (b) ( )

32

01

kk

=

+∑ (c) ( )0

22 1

jj

=−

+∑

(d)3

12i

i=−∑ (e) ( )

22 1

01 k

k

+

=

−∑ (f) ( )3

1log 10i

i=∑

(g)4

1 1n

nn= +∑ (h)

( )

( )

42

24

2

2

1

1

i

i

i

i

=

=

+

+

∑

∑ (i)

3

0

12256k

k=∑

2. Whichofthefollowingisthevalueof ( )4

04 1

kk

=

+∑ ?

(1)53 (3)37

(2)45 (4)80

3. Thesum7

7

42i

i

−

=∑ isequalto

(1)158 (3) 34

(2) 32 (4) 78


Ms.Talhami 13

4. Writeeachofthefollowingsumsusingsigmanotation.Usekasyourindexvariable.Note,therearemanycorrectwaystowriteeachsum(andevenmoreincorrectways).

(a) 2 4 8 64 128− + + − + ⋅ ⋅ ⋅+ + − (b)1 1 1 1 11 4 9 81 100+ + + ⋅⋅⋅+ + (c) 4 9 14 44 49+ + + ⋅ ⋅ ⋅+ +

5. Whichofthefollowingrepresentsthesum 2 5 10 82 101+ + + ⋅ ⋅ ⋅+ + ?

(1) ( )6

14 3

jj

=

−∑ (3) ( )10

2

11

jj

=

+∑

(2) ( )103

32

jj

=

−∑ (4) ( )11

04 1j

j=+∑

6. Asequenceisdefinedrecursivelybytheformula 1 2 1 24 2 with 1 and 3n n nb b b b b− −= − = = .Whatisthevalue

of5

3i

ib

=∑ ?Showtheworkthatleadstoyouranswer.

REASONING

6. Acuriouspatternoccurswhenwelookatthebehaviorofthesum ( )12 1

n

kk

=

−∑ .

(a) Findthevalueofthissumforavarietyofvaluesofnbelow:

( )2

12 : 2 1

kn k

=

= − =∑ ( )4

14 : 2 1

kn k

=

= − =∑

( )3

13 : 2 1

kn k

=

= − =∑ ( )5

15 : 2 1

kn k

=

= − =∑

(b)What types of numbers are you summing?Whattypesofnumbersarethesums?

(c) Findthevalueofnsuchthat .


Ms.Talhami 14

ARITHMETIC SERIES COMMONCOREALGEBRAII

Aseriesissimplythesumofthetermsofasequence.Thefundamentaldefinition/notionofaseriesisbelow.Intruth,youhavealreadyworkedextensivelywithseriesinpreviouslessonsalmostanytimeyouevaluatedasummationproblem.

Exercise#1:Giventhearithmeticsequencedefinedby 1 12 and 5n na a a −= − = + ,thenwhichofthefollowingis

thevalueof5

51

ii

S a=

=∑ ?

(1)32 (3)25

(2)40 (4)27Thesumsassociatedwitharithmeticsequences,knownasarithmeticseries,haveinterestingproperties,manyapplicationsandvaluesthatcanbepredictedwithwhatiscommonlyknownasrainbowaddition.

Exercise#2:Considerthearithmeticsequencedefinedby 1 13 and 2n na a a −= = + .Theseries,basedonthefirsteighttermsofthissequence,isshownbelow.Termshavebeenpairedoffasshown.

(a) Whatdoeseachofthepairedoffsumsequal?(b)Whydoesitmakesensethatthissumisconstant?(c) Howmanyofthesepairsarethere?(e) Generalizethisnowandcreateaformulaforanarithmeticseriessumbasedonlyonitsfirstterm, 1a ,itslast

term, na ,andthenumberofterms,n.

THEDEFINITIONOFASERIES

Iftheset representtheelementsofasequencethentheseries, ,isdefinedby:

(d)Usingyouranswersto(a)and(c)findthevalueofthesumusingamultiplicativeprocess.


Ms.Talhami 15

Exercise#3:Whichofthefollowingisthesumofthefirst100naturalnumbers?Showtheprocessthatleadstoyourchoice.

(1)5,000 (3)10,000

(2)5,100 (4)5,050Exercise#4:Findthesumofeacharithmeticseriesdescribedorshownbelow.Exercise#5:Kirkhassetupacollegesavingsaccountforhisson,Maxwell.IfKirkdeposits$100permonthinanaccount,increasingtheamounthedepositsby$10permontheachmonth,thenhowmuchwillbeintheaccountafter10years?

SUMOFANARITHMETICSERIES

Givenanarithmeticserieswithnterms, ,thenitssumisgivenby:

(a) Thesumofthesixteentermsgivenby:

.

(b) Thefirsttermis ,thecommondifference,d,is6andthereare20terms

(c) The last term is and the commondifference,d,is .

(d) Thesum .


Ms.Talhami 16

ARITHMETIC SERIES COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Whichofthefollowingrepresentsthesumof3 10 87 94+ + ⋅ ⋅ ⋅+ + ifthearithmeticserieshas14terms?

(1)1,358 (3)679 (2)658 (4)1,2762. Thesumofthefirst50naturalnumbersis

(1)1,275 (3)1,250 (2)1,875 (4)9503. Ifthefirstandlasttermsofanarithmeticseriesare5and27,respectively,andtheserieshasasum192,

thenthenumberoftermsintheseriesis

(1)18 (3)14 (2)11 (4)124. Findthesumofeacharithmeticseriesdescribedorshownbelow.

(a) The sum of the first 100 even, naturalnumbers.

(b) The sum of multiples of five from 10 to 75,inclusive.

(c) Aserieswhosefirsttwotermsare,respectively,andwhoselasttermis124.

(d) Aseriesof20termswhoselasttermisequalto97andwhosecommondifferenceisfive.


Ms.Talhami 17

5. Foranarithmeticseriesthatsumsto1,485,itisknownthatthefirsttermequals6andthelasttermequals93.Algebraicallydeterminethenumberoftermssummedinthisseries.

APPLICATIONS6. ArlingtonHighSchoolrecentlyinstalledanewblack-boxtheatreforlocalproductions.Theyonlyhadroom

for14rowsofseats,wherethenumberofseatsineachrowconstitutesanarithmeticsequencestartingwitheightseatsandincreasingbytwoseatsperrowthereafter.Howmanyseatsareinthenewblack-boxtheatre?Showthecalculationsthatleadtoyouranswer.

7. Simeon starts a retirement account where he will place $50 into the account on the first month and

increasinghisdepositby$5permontheachmonthafter.Ifhesavesthiswayforthenext20years,howmuchwilltheaccountcontaininprincipal?

8. The distance an object falls per second while only under the influence of gravity forms an arithmetic

sequencewithitfalling16feetinthefirstsecond,48feetinthesecond,80feetinthethird,etcetera.Whatisthetotaldistanceanobjectwillfallin10seconds?Showtheworkthatleadstoyouranswer.

9. Alargegrandfatherclockstrikesitsbellonceat1:00,twiceat2:00,threetimesat3:00,etcetera.Whatis

thetotalnumberoftimesthebellwillbestruckinaday?Useanarithmeticseriestohelpsolvetheproblemandshowhowyouarrivedatyouranswer.


Ms.Talhami 18

GEOMETRIC SERIES COMMONCOREALGEBRAII

Justaswecansumthetermsofanarithmeticsequencetogenerateanarithmeticseries,wecanalsosumthetermsofageometricsequencetogenerateageometricseries.

Exercise#1: Givenageometric seriesdefinedby the recursive formula 1 13 and 2n na a a −= = ⋅ ,whichof the

followingisthevalueof5

51

ii

S a=

=∑ ?

(1)106 (3)93 (2)75 (4)35Thesumofafinitenumberofgeometricsequencetermsislessobviousthanthatforanarithmeticseries,butcanbefoundnonetheless.Thenextexercisederivestheformulaforfindingthissum.Exercise#2:Recallthatforageometricsequence,thenthtermisgivenby 1

1n

na a r −= ⋅ .Thus,thegeneralformofangeometricseriesisgivenbelow.

2 2 11 1 1 1 1

n nnS a a r a r a r a r− −= + + + ⋅⋅⋅+ +

Exercise#3:Whichofthefollowingrepresentsthesumofageometricserieswith8termswhosefirsttermis3andwhosecommonratiois4? (1)32,756 (3)42,560 (2)28,765 (4)65,535

(a) Writeanexpressionbelowfortheproductofrand .

(b) Find,insimplestform,thevalueof intermsof .

(c) Writebothsidesoftheequationin(b)intheirfactoredform.

(d) Fromtheequationinpart(c),findaformulaforintermsof .


Ms.Talhami 19

Exercise#4:Findthevalueofthegeometricseriesshownbelow.Showthecalculationsthatleadtoyourfinalanswer.

6 12 24 768+ + + ⋅ ⋅ ⋅+

Exercise #5: Maria places $500 at the beginning of each year into an account that earns 5% interestcompoundedannually.Mariawouldliketodeterminehowmuchmoneyisinheraccountaftershehasmadeher$500depositattheendof10years.Exercise#6:Apersonplaces1pennyinapiggybankonthefirstdayofthemonth,2penniesonthesecondday,4penniesonthethird,andsoon.Willthispersonbeamillionaireattheendofa31daymonth?Showthecalculationsthatleadtoyouranswer.

SUMOFAFINITEGEOMETRICSERIES

Forageometricseriesdefinedbyitsfirstterm, ,anditscommonratio,r,thesumofntermsisgivenby:

or

(a) Determineaformulafortheamount, ,thatagiven$500hasgrown to t-yearsafter itwasplacedintothisaccount.

(b) At the end of 10 years, which will be worthmore:the$500investedinthefirstyearorthefourthyear?Explainbyshowinghowmucheachisworthatthebeginningofthe11thyear.

(c) Based on (b), write a geometric sumrepresenting the amount ofmoney inMaria’saccountafter10years.

(d) Evaluatethesumin(c)usingtheformulaabove.


Ms.Talhami 20

GEOMETRIC SERIES COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Findthesumsofgeometricserieswiththefollowingproperties:

(a) 1 6, 3 and 8a r n= = = (b) 1120, , and 62a r n= = = (c) 1 5, 2, and 10a r n= − = − =

2. Ifthegeometricseries 12854 3627

+ + ⋅⋅⋅+ hasseventermsinitssumthenthevalueofthesumis

(1) 411827

(3)13709

(2)12743

(4) 824154

3. Ageometricserieshasafirsttermof32andafinaltermof 14

− andacommonratioof 12

− .Thevalueof

thisseriesis

(1)19.75 (3)22.5 (2)16.25 (4)21.25

4. Whichofthefollowingrepresentsthevalueof8

0

32562

i

i=

⎛ ⎞⎜ ⎟⎝ ⎠

∑ ?Thinkcarefullyabouthowmanytermsthis

serieshasinit.

(1)19,171 (3)22,341 (2)12,610 (4)8,9565. Ageometricserieswhosefirsttermis3andwhosecommonratiois4sumsto4095.Thenumberofterms

inthissumis (1)8 (3)6 (2)5 (4)4


Ms.Talhami 21

6. Findthesumofthegeometricseriesshownbelow.Showtheworkthatleadstoyouranswer.

127 9 3729

+ + + ⋅⋅⋅

APPLICATIONS

7. Inthepictureshownattheright,theoutermostsquarehasanareaof16squareinches.Allothersquaresareconstructedbyconnectingthemidpointsofthesidesofthesquareitisinscribedwithin.Findthesumoftheareasofallofthesquaresshown.First,considerthehowtheareaofeachsquarerelatestothelargersquarethatsurrounds(circumscribes)it.

8. Acollegesavingsaccountisconstructedsothat$1000isplacedtheaccountonJanuary1stofeachyearwith

aguaranteed3%yearlyreturnininterest,appliedattheendofeachyeartothebalanceintheaccount.Ifthisisrepeatedlydone,howmuchmoneyisintheaccountafterthe$1000isdepositedatthebeginningofthe19thyear?Showthesumthatleadstoyouransweraswellasrelevantcalculations.

9. Aballisdroppedfrom16feetaboveahardsurface.Aftereachtimeithitsthesurface,itreboundstoa

heightthatis 34 ofitspreviousmaximumheight.Whatisthetotalverticaldistance,tothenearestfoot,

theballhastraveledwhenitstrikesthegroundforthe10thtime?Writeoutthefirstfivetermsofthissumtohelpvisualize.


Ms.Talhami 22

MORTGAGE PAYMENTS COMMONCOREALGEBRAII

Mortgages, not just on houses, are large amounts of money borrowed from a bank on which interest iscalculated(addedon)onaregular(typicallymonthly)basis.Regularpaymentsarealsomadeontheamountofmoneyowedsothatovertimetheprincipal(originalamountborrowed)ispaidoffaswellasanyinterestontheamountowed.Thisisacomplexprocessthatultimatelyinvolvesgeometricseries.First,somebasics.Exercise#1:Let'ssayapersontakesoutamortgagefor$200,000andwantstomakepaymentsof$1600eachmonthofpayitoff.Thebankisgoingtochargethisperson4%nominalyearlyinterest,appliedmonthly.(a) Whatistheamountowedattheendofthefirstmonth?Showthecalculationsthatleadtoyouranswer.(b) Howmuchofthefirstmonth'spaymentwenttopayingofftheprincipal?Howmuchofitwenttopaying

interestontheloan?Showyourcalculations.

AmountTowardsPrincipal AmountTowardsInterest(c) Determinetheamountowedattheendofthesecondmonth.Again,showthecalculationsthatleadtoyour

answer.(d) The amount owed at the endof amonth actually forms a sequence that can be defined recursively. If

1 200,000a = ,thendefinearecursiverulethatgivesthissequence.Exercise#2:Ifapersontookouta$150,000mortgageat5%yearlyinterest,whywoulditbeunwisetohavemonthlypaymentsof$500?


Ms.Talhami 23

Now,evenifwecandefinethesequencerecursively,asinExercise#1(d),itwouldbenicetohaveaformulathatwouldcalculatewhatweowedafteracertainnumberofmonthsexplicitly.Todothis,wemustseeatricky,extendedpattern.

Exercise#3:Let'sgobacktoourexampleofthe$200,000mortgageat4%yearlyinterest.Remember,wearepayingoffthismortgagewith$1,600monthlypayments(muchofwhichareinitiallygoingtointerest).Let'sseeifwecandeterminehowmuchwestilloweaftern-payments.Tomakeourworkeasiertofollow(andmore

general),wewilllet .04 .00312

r = = , 200,000P = ,and 1,600m = tostandformonthlyrate(indecimalform,our

principal,andourmonthlypayment).

Exercise#4:Nowlet'sseethegeometricseries.Inyouranswerto(d),youshouldhavethefollowingexpression:

( ) ( ) ( ) ( ) ( )1 2 3 21 1 1 1 1n n nm r m r m r m r m r m− − −− + − + − + −⋅⋅⋅− + − + − Given that this is equivalent to: ( ) ( ) ( ) ( )( )2 3 11 1 1 1 nm m r m r m r m r −− + + + + + + + ⋅⋅⋅ + , find the sum of the

geometricseriesinsideoftheparentheses.Thinkcarefullyaboutthenumberoftermsinthisexpression.

Exercise#5:Findtheamountowedonthisloanafter10yearsor120payments(months).

(a) Explain what the following calculationrepresents.

(b)Whatdoesthefollowingcalculationrepresent?Write it in expanded form, but leave thebinomial .

(c) Based on (a) and (b), continue this line ofthinking to write expressions for the amountowedattheendof3monthsand4months.

(d) Basedon (c),writeanequation forhowmuchwouldbeowedafternmonths.


Ms.Talhami 24

MORTGAGE PAYMENTS COMMONCOREALGEBRAIIHOMEWORK

APPLICATIONS1. Considerthemortgageloanof$150,000atanominal6%yearlyinterestappliedmonthlyatarateof0.5%

permonth.Monthlypaymentsof$1,000arebeingmadeonthisloan. (a) Determinehowmuchisowedonthisloanattheendofthefirst,second,thirdmonthandfourthmonths.

Showtheworkthatleadstoyouranswers.Evaluateallexpressions. OneMonth: TwoMonths: ThreeMonths: FourMonths: (b) Theamountsthatareowedattheendofeachmonthformasequencethatcanbedefinedrecursively.

Giventhat 1 $150,000a = representsthefirstamountowed,givearecursiverulebasedonwhatyoudidin(a)thatshowshoweachsuccessiveamountoweddependsonthepreviousone.

(c) Usingageometricseriesapproach(i.e.theformulawedevelopedinExercises#3and#4),determine

howmuchisstillowedafter5yearsofpayments.Showyourwork. (d)Willthisloanbepaidoffafter20years?Whatabout30?Provideevidencetosupportbothanswers.


Ms.Talhami 25

Whenaloanofficerspeakstopeopleaboutaloanforacertainprincipal,P,atacertainmonthlyrate,r,theyalwayshavetobalancetwoquantities,themonthlypayment,m,withthenumberofpayments,n,ittakestopayofftheloan.Thesetwovaryinversely.Allofthesequantitiescanberelatedbytheformula:

( )1 1 nP rmr −

⋅=− +

ThisformulaisderivedbytakingtheformulawearrivedatinExercises#3and#4andsettingwhatweoweequaltozero.2. Calculate themonthlypaymentneeded topayoffa$200,000 loanat4%yearly interestovera20year

period.Recallthatristhemonthlyrate.Showyourworkandcarefullyevaluatetheaboveformulaform.3. Dothesamecalculationasinthepreviousexercisebutnowmakethepayoffperiod30yearsinsteadof20.

Howmuchlessisyourmonthlypayment?Itisofinteresttoalsobeabletocalculatethenumberofpayments(andhencethepayoffperiod)ifyouhaveamonthlypaymentinmind.Butthisismuchmoredifficultgiventhatyoumustsolveforn.4. Giventheformulaabove:

(a)Showthat( )

log 1

log 1

P rmnr

⋅⎛ ⎞− −⎜ ⎟⎝ ⎠=+

(b)Using(a),determinethenumberofmonthsitwould take to pay off a $150,000 loan at amonthly0.5%ratewith$1,000payments.

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