Sec 5.8 – Sec 6.2 – (Arithmeti

CarlFriedrichGaussisprinhistory.Asthestorygbrillianceattheageof8causingdisruptionsinclbetween1and100.

1+2+3+4+5+

TheteacherexpectedtGaussdiditinseconds.So,theteacherThistimeGaussdidn’tevenmove,hejunumbersbypairingtheminaspecialw

1+2+3+4+5

Hedeterminedthatifyoufindthesumofthafterthatsumsto101andsoon.Inshort,t

1 2 3 4 5 6

UsingthetechniqethatGaussmayhavedev

1 2 3 4 5 6

Itturnsoutthatthisstrategyworksforthe

Determinethesumofthefollowingpartia

1.2 4 6 8 ……… . 116 118

The Sum of ‘n’ terms of an

arithmetic series

The represents the number of pairs of the

terms that form the special sum.

repreterm

Mathematical Modeling ic & Geometric Series)        Nam

robablyoneofthemostnotedcompletemathemagoes,hewaspotentiallyreconginizedforhismathwhenhewasassignedbusyworkbyhisteacherflass.Hewastoldbytheteachertoaddallofthen

+6+…………..+97+98+99+100=

thistasktotakeGuassseveralminutestoanhourthinkinghehadcheatedtoldhimtoaddthenumustrepondedwiththeanswer.Hehaddevisedatwayattheageof8.Howdidhedoit?

+…………+96+97+98+99+100=

hemostouterpairofnumbersitsumsto101andthatthereshouldbe50pairsofnumbersthatsumsto101.

………… . 96 97 98 99 100

veloped,determinethesumofalltheintegersfrom1t

………… . 196 197 198 199 200

partialsumofanyArithmeticSeries.Considerwriting

alarithmeticseriesusingtheformula.

120 2.FindtheS62ofthefollowing4 9 14 19 ………

M.WinkingUnit6‐2page107

101

101101

101

101

The “a1” esents the first

m of the series.

The “an” represents the last term of the series.

51 pairs o

me:         

aticianshematicalfornumbers

rtokeephimbusybutbersbetween1and200.ricktoaddconsecutive

thenextinnerpairSo,thissuggests:

5151

to200.

gitasaformula.

gseries:….

of 101

Determinethesumofthefollowingpartialarithmeticseriesusingtheformula.

3.30 26 22 ……… 102 106

4.FindtheS42,giventhata1=6anda42=129

5.FindtheS39giventhata1=6andd=6

6.FindtheS34giventhata34=73andd=2.

7.Determinethevalueof

8

1

32n

n

8.Determinethevalueof

42

1

228n

n

9.Addisondecidestotrytosavemoneyinajarathome.Shedecidestosave$20thefirstweekoftheyearandeachweekshewillincreasetheamountshesavesby$5.So,onthesecondweekshewillsave$25andthenonthirdweekshewillsaveanadditional$30.Thisprocesswouldrepeatforthewholeyearof52weeks.Howmuchmoneyshouldshehaveinthejarattheendoftheyear?

M.WinkingUnit6‐2page108

TherearealsoformulasthatcanbecreatedtofindthesumofaGeometricSeries.Firstconsiderthefollowingseries.

3 6 12 24 48 96 192 384 768 1536 3072 Thiscouldalsobere‐writtenas: 3 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2

So,anygeometricseriescouldbewrittenas:

………… . . Considermultiplyingbothsidesbya“ ”

∙ ………… . . Next,addthetwoseriessimilartohowyouuseeliminationinsolvingasystemofequations.

………… . . ∙ ………… . .

ThisformulaworksforthepartialsumofanyGeometricSeries.

Determinethesumofthefollowingpartialgeometricseriesusingtheformula.

1.FindtheS14ofthefollowingseries:2 6 18 54 162 ……….

2.3 6 12 24 …… 98304 196608

1st term 2nd term 3rd term 4th term 5th term 6th term 7th term 8th term 9th term 10th term 11th term

+

The Sum of ‘n’ terms of an

arithmetic series

The “n” represents the number of sequential terms to

be included in the sum.

The “a1” represents the first term of the series.

The “r” represents the common ratio from one

term to the next.

1

1

1

2

n

n n

a a n d

nS a a

11

1 1

1

nn

n

n

a a r

a rS

r

2

1

ar

a

2 1d a a

Arithmetic

Geometric

M.WinkingUnit6‐2page109

Determinethesumofthefollowingpartialgeometricseriesusingtheformula.

3.Determinethesumofthefirst11terms(S11)forageometricseriesgiventhefirsttermis6(a1=6)andthecommonratiois5(r=5).

4.Giventhesumofthefirst12termsofageometricsequencessumto20475andthecommonratiois2(r=3),determinethefirstterm(a1).

7.Determinethevalueof

5

1

42n

n

8.Determinethevalueof

9.72 36 18 9 4.5 …… . .

10. …… . .

11.Determinethevalueofn

n

1 2

112

12.Determinethevalueofn

n

1 2

34

1

1

1

2

n

n n

a a n d

nS a a

11

1 1

1

nn

n

n

a a r

a rS

r

2

1

ar

a

2 1d a a

Arithmetic

Geometric

M.WinkingUnit6‐2page110

UsingtheAlgebraortheInfiniteGeometric

13.0.5555555555

13.14.14141414

15.0.99999999

16.Kellydecidestostartsavingmoney.O($0.01).Then,foreachweekthatfollopreviousweek.So,onthesecondweekweek4cents($0.04).Ifthisprocesswweeks,howmuchmoneywouldKellyh

16.SarahPinkskiwascreatingapatternherpatterngrows.Shehasalreadyusemanytileswouldittakeintotaltocrea

Seriesformulasdeterminethefractionforthefo

14.0.3434343434

14.0.450450450450

Onthefirstweekoftheyear,shesavesonecentowsshecontinuestodoubletheamountshesavedkshesavedanadditional2cents($0.02)andthewereabletobecontinuedfortheentireyearof52havesavedbytheendoftheyear?

usingtriangletiles.Shewantedtoshoweachsuced40triangulartilestocreatethepatternbelow.ate10stepsofthedesign?

M.WinkingUnit6‐2page111

llowingrepeatingdecimals.

dthe3rd

ccessivesteptoshowhowIfshecontinuedhow