14.2 Arithmetic and Geometric Sequences 1

14.2ArithmeticandGeometricSequences 1

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Twocommonsequencesthatappearfrequentlyinmathematicsarethearithmeticandgeometricsequences.

ArithmeticSequenceAnarithmeticsequenceisoneinwhichthesamenumberisaddedorsubtractedfromeachtermtogetthenextterminthesequence.Thenumberyouaddorsubtractiscalledthecommondifference.

Handyrulesinvolvinganarithmeticsequence:

𝑎! = 𝑎! 𝑎! = 𝑎! + 𝑑 𝑎! = 𝑎! + 2𝑑𝑎! = 𝑎! + 3𝑑𝑎! = 𝑎! + 4𝑑

⋮

NthtermofanArithmeticSequenceThenthtermofanarithmeticsequencewithfirstterma1andcommondifferencedisgivenby:

𝑎! = 𝑎! + 𝑛 − 1 𝑑

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:

𝑆! =𝑛(𝑎! + 𝑎!)

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:

𝑎! = 𝑎!𝑟!!!

14.2ArithmeticandGeometricSequences 3

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

SumofaFiniteGeometricSeriesThenthtermofageometricsequencewithfirstterma1andcommonratiorisgivenby:

𝑆! = 𝑎!1− 𝑟!

1− 𝑟

Practice 14.2

For each sequence, state if it is arithmetic, geometric, or neither.If it is arithmetic, tell thecommon difference. If it is geometric, tell the common ratio. If it is neither, chill out and moveon to the next problem.

1) -1, 6, -36, 216, -1296, ... 2) 11, -9, -29, -49, -69, ...

3) 2, 52

, 3, 72

, 4, ... 4) -6, 24, -126, 624, -3126, ...

5) 32, 36, 40, 44, 48, ... 6) 0.4, 2, 10, 50, 250, ...

7) an = -

1924

+ 53n 8) a

n = 8 + 6n

9) an = 3 × (-6)n - 1

10) an =

2n2n + 1

Determine if the sequence is arithmetic. If it is, find the common difference, the term named inthe problem, and the explicit formula.

11) 10, 16, 22, 28, ...Find a

25

12) -31, -33, -35, -37, ...Find a

35

13) 1, 2, 6, 24, ...Find a

20

Determine if the sequence is geometric. If it is, find the common ratio, the term named in theproblem, and the explicit formula.

14) 1, 4, 16, 64, ...Find a

9

15) -7, -5, -2, 2, ...Find a

10

16) 1, -2, 4, -8, ...Find a

10

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Fornumbers16–20,findthesumofthefirstntermsindicatedinpart(a).Then,forpart(b),findnforthegivensumSn.

17. 1+4+16+64+…. 18. 50+42+34+26+… 19. 7+(-21)+63+(-189)+…

a. Sumofthefirst14terms? a. Sumofthefirst40terms? a. Sumofthefirst84terms?b. ForwhichtermwouldSn=341? b. ForwhichtermwouldSn=182? b. ForwhichtermwouldSn=3829?

20. 2+16+30+44+58+… 21. 1+9+81+729+…. 22. 3+8+13+18+23+…

a. Sumofthefirst24terms? a. Sumofthefirst10terms? a. Sumofthefirst20terms?b. ForwhichtermwouldSn=2178 b. ForwhichtermwouldSn=820? b. ForwhichtermwouldSn=366?

Evaluateeachseries.

23. 24. 25.

Writeeachseriesinsigmanotation.26. 16+25+36+49+64 27. 2+4+8+16+32 28. 501+502+503+504

Skillz Review! Writetheequationofalinewiththegivenslopethatpassesthroughthegivenpoint.Inslope-interceptform:y=mx+b Inpoint-slopeform:y-y1=m(x–x1)1. slope=-3; through(-1,3) 2. slope=0;through(-2,3) 3. slope=3; through(1,-3) 4. slope=− 𝟑

𝟓;through(0,0)

! 𝑖 + 2!

!!!

! 𝑗!!

!!!

! 𝑡!!

!!!

14.2ArithmeticandGeometricSequences Application5

Write your questionsthoughts here! 1. Givenoneexampleofasequencethatwouldbearithmeticandoneexamplethatwouldbegeometric.

2. Findthesumofthefirst18termsofthearithmeticseries1+5+9+13+…

3. Nextyear,theAlgebros’MarchMathnesstournamentwillbebiggerthanever!Inthefirstround,64gameswillbeplayed.Ineachsuccessiveround,thenumberofmatchesplayeddecreasesbyonehalf.

a. Findaruleforthenumberofgamesplayedinthenthround.Forwhatvaluesofndoesyourrulemakesense?

b. Findthetotalnumberofgamesthatwillbeplayed.

4. TheSierpinskitriangleisadesignusingequilateraltriangles.Theprocessinvolvesremovingsmallertrianglesfromlargertrianglesbyjoiningthemidpointsofthesidesofthelargertrianglesasshownbelow.Assumethattheinitialtriangleisequilateralwithsides1unitlong.

a. Letanbethenumberoftrianglesremovedatthenthstage.Findaruleforan.Thenfindthetotalnumberoftrianglesremovedthroughthe10thstage.

b. Letbnbetheremainingareaoftheoriginaltriangleatthenthstage.Findaruleforbn.Thenfindtheremainingareaoftheoriginaltriangleatthe15thstage.