14.2ArithmeticandGeometricSequences 1

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Twocommonsequencesthatappearfrequentlyinmathematicsarethearithmeticandgeometricsequences.

ArithmeticSequenceAnarithmeticsequenceisoneinwhichthesamenumberisaddedorsubtractedfromeachtermtogetthenextterminthesequence.Thenumberyouaddorsubtractiscalledthecommondifference.

Handyrulesinvolvinganarithmeticsequence:

!! = !! !! = !! + ! !! = !! + 2!!! = !! + 3!!! = !! + 4!

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NthtermofanArithmeticSequenceThenthtermofanarithmeticsequencewithfirstterma1andcommondifferencedisgivenby:

!! = !! + ! − 1 !

Example1.Arethefollowingarithmeticsequences? 3,8,13,18,23,28... -2,-12,-22,-32,… 2,4,8,16,32,… 14,14.5,15,15.5,16… Example2.Giventwotermsinanarithmeticsequence,findthecommondifference,the52ndterm,andtheexplicitformula. a20

=70 a33=96

Pre-Calculus

14.2ArithmeticandGeometricSequences 2

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Time-outofforsomemathhistory….

SumofaFiniteArithmeticSequenceThesumofthefirstntermsofanarithmeticsequenceisgivenby:

!! =!(!! + !!)

2

Example3:Findthesumofthefirst20termsofthearithmeticseries:

a.2+6+10+14+18+...b.Supposethesumoftheserieshasasumof2178.FindnsuchthatSn=2178.

GeometricSequenceAgeometricsequenceisoneinwhichthesamenumberismultipliedordividedbyeachtermtogetthenextterminthesequence.Thenumberyoumultiplyordividebyiscalledthecommonratio,usuallydenotedbyr.Determineifthefollowingsequencesarearithmetic,geometricorneither.4.1,2,6,24,120,…5.81,27,9,3,1,… 6. 5,10,15,20,25,…

Example7:Thethirdtermofageometricseriesequals64whilethecommonratiois2.

a. Writearuleforthenthterm. b.Findthe9thterm

NthtermofaGeometricSequenceThenthtermofageometricsequencewithfirstterma1andcommonratiorisgivenby:

!! = !!!!!!

14.2ArithmeticandGeometricSequences 3

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Example8:Findthesumofthefirst10termsoftheseries1+5+25+125+…Example9.ForwhichtermwouldSn=3906?

SumofaFiniteGeometricSeriesThenthtermofageometricsequencewithfirstterma1andcommonratiorisgivenby:

!! = !!1− !!1− !