14.2 Arithmetic and Geometric Sequences 1 Write your questions and thoughts here! Two common sequences that appear frequently in mathematics are the arithmetic and geometric sequences. Arithmetic Sequence An arithmetic sequence is one in which the same number is added or subtracted from each term to get the next term in the sequence. The number you add or subtract is called the common difference. Handy rules involving an arithmetic sequence: ! ! = ! ! ! ! = ! ! + ! ! ! = ! ! + 2! ! ! = ! ! + 3! ! ! = ! ! + 4! ⋮ Nth term of an Arithmetic Sequence The nth term of an arithmetic sequence with first term a 1 and common difference d is given by: ! ! = ! ! + ! − 1 ! Example 1. Are the following arithmetic sequences? 3, 8, 13, 18, 23, 28... -2, -12, -22, -32, … 2, 4, 8, 16, 32, … 14, 14.5, 15, 15.5, 16… Example 2. Given two terms in an arithmetic sequence, find the common difference, the 52nd term, and the explicit formula. a 20 = 70 a 33 = 96 Pre-Calculus
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14.2ArithmeticandGeometricSequences 1
Write your questions and thoughts here!
Twocommonsequencesthatappearfrequentlyinmathematicsarethearithmeticandgeometricsequences.
ArithmeticSequenceAnarithmeticsequenceisoneinwhichthesamenumberisaddedorsubtractedfromeachtermtogetthenextterminthesequence.Thenumberyouaddorsubtractiscalledthecommondifference.
Handyrulesinvolvinganarithmeticsequence:
!! = !! !! = !! + ! !! = !! + 2!!! = !! + 3!!! = !! + 4!
⋮
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− !
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