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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
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