O SlideShare utiliza cookies para otimizar a funcionalidade e o desempenho do site, assim como para apresentar publicidade mais relevante aos nossos usuários. Se você continuar a navegar o site, você aceita o uso de cookies. Leia nosso Contrato do Usuário e nossa Política de Privacidade.
O SlideShare utiliza cookies para otimizar a funcionalidade e o desempenho do site, assim como para apresentar publicidade mais relevante aos nossos usuários. Se você continuar a utilizar o site, você aceita o uso de cookies. Leia nossa Política de Privacidade e nosso Contrato do Usuário para obter mais detalhes.
Sequences andSequences andSeriesSeriesAlgebra 2 – Ironwood High SchoolAlgebra 2 – Ironwood High School
SequencesSequences & Series& Series SequenceSequence An ordered list ofAn ordered list ofnumbersnumbers A progression ofA progression ofnumbersnumbers Can be arithmetic,Can be arithmetic,geometric or neithergeometric or neither Can be finite orCan be finite orinfiniteinfinite SeriesSeries A value you get whenA value you get whenyou add up the termsyou add up the termsof a sequencesof a sequences Sum of numbers in aSum of numbers in asequencesequence Uses summationUses summationnotationnotation ΣΣ (sigma)(sigma)
Try this…Try this…Find the 10Find the 10ththterm of this sequenceterm of this sequence2, 5, 8,…2, 5, 8,… Start by determining the patternStart by determining the pattern Adding 3 to the previous numberAdding 3 to the previous number Known as a common differenceKnown as a common difference Continue the pattern to get the 10Continue the pattern to get the 10ththtermterm 2, 5, 8, 11, 14, 17, 20, 23, 26,…2, 5, 8, 11, 14, 17, 20, 23, 26,… So, the 10So, the 10ththterm is 29term is 29
Try This…Try This…Write the first 7 terms ofWrite the first 7 terms of aann = 4= 4nn + 9+ 9aa11 = 13= 13aa22 = 17= 17aa33 = 21= 21aa44 = 25= 25aa55 = 29= 29aa66 = 33= 33aa77 = 37= 37
Determining Rules forDetermining Rules fora Sequencea SequenceExample:Example:Determine a rule for theDetermine a rule for the nnth term of theth term of thesequence: 1, 16, 81, 256,. . .sequence: 1, 16, 81, 256,. . . When determining a rule for a sequence you need to compare theWhen determining a rule for a sequence you need to compare theterm number to the actual term.term number to the actual term. For this sequence 81 is the 3For this sequence 81 is the 3rdrdterm, so you need to determine how to get 81term, so you need to determine how to get 81from 3.from 3.Rule: aRule: ann = 4^n= 4^n
Determining if aDetermining if asequence is Geometric,sequence is Geometric,Arithmetic or NeitherArithmetic or Neither Watch the following video on ArithmeticWatch the following video on Arithmeticand Geometric Sequencesand Geometric Sequences http://www.virtualnerd.com/algebra-2/sequenhttp://www.virtualnerd.com/algebra-2/sequen
Determine if theDetermine if thefollowing sequences arefollowing sequences areArithmetic, Geometric, orArithmetic, Geometric, orNeitherNeitherProblems:Problems:1.1. 3, 8, 13, 18, 23,…3, 8, 13, 18, 23,…2.2. 1, 2, 4, 8, 16,…1, 2, 4, 8, 16,…3.3. 24, 12, 6, 3, 3/2,24, 12, 6, 3, 3/2,3/4,…3/4,…4.4. 55, 51, 47, 43, 39,55, 51, 47, 43, 39,35,…35,…5.5. 2, 5, 10, 17,…2, 5, 10, 17,…6.6. 1, 4, 9, 16, 25, 36,1, 4, 9, 16, 25, 36,……Answers:Answers:1.1. Arithmetic, the commonArithmetic, the commondifference is 5.difference is 5.2.2. Geometric, theGeometric, thecommon ratio is 2.common ratio is 2.3.3. Geometric, theGeometric, thecommon ratio is ½.common ratio is ½.4.4. Arithmetic, the commonArithmetic, the commondifference is -4.difference is -4.5.5. Neither, no commonNeither, no commonratio or difference.ratio or difference.6.6. Neither, no commonNeither, no commonratio or difference.ratio or difference.
Infinite Vs. FiniteInfinite Vs. Finite InfiniteInfinite A sequence that goesA sequence that goeson foreveron foreverExample:Example:14, 28, 42, 56, 70,…14, 28, 42, 56, 70,… FiniteFinite A sequence that hasA sequence that hasan endan endExample:Example:1, 3, 9, 27, and 81.1, 3, 9, 27, and 81.
Arithmetic RuleArithmetic Ruleaann = a= a11 + (n - 1)d+ (n - 1)d aa11 is the first term in the sequenceis the first term in the sequence n is the number of the term you aren is the number of the term you aretrying to determinetrying to determine d is the common differenced is the common difference aann is the value of the term that areis the value of the term that arelooking forlooking for
Try this….Try this….Use the arithmetic formula to determine the 100Use the arithmetic formula to determine the 100ththterm ofterm ofthe following sequence:the following sequence:75, 25, -25, -75, -125,…75, 25, -25, -75, -125,… aa11 = 75= 75 n = 100n = 100 d = -50d = -50aann == aa11 + (+ (nn - 1)- 1)dd== 7575 + (+ (100100 – 1)(– 1)(-50-50))= -4875= -4875
Example of a ArithmeticExample of a ArithmeticSequence in the RealSequence in the RealWorldWorldSuppose you are training to run a 6 mileSuppose you are training to run a 6 milerace. You plan to start your training byrace. You plan to start your training byrunning 2 miles a week, and then yourunning 2 miles a week, and then youplan to add a ½ mile more every week.plan to add a ½ mile more every week.At what week will you be running 6At what week will you be running 6miles?miles?
SolutionSolution The first term of the sequence will be the initial numberThe first term of the sequence will be the initial numberof miles you plan on running.of miles you plan on running. The common difference of the sequence will be the ½The common difference of the sequence will be the ½mile that you increase every week.mile that you increase every week. n will stand for the number of weeks it will take you ton will stand for the number of weeks it will take you toreach 6 miles.reach 6 miles.aann == aa11 + (n - 1)+ (n - 1)dd66 == 22 + (n – 1)(+ (n – 1)(1/21/2))
Geometric RuleGeometric Ruleaann = a= a11*r*r(n-1)(n-1) aa11 is the 1is the 1ststterm of the sequenceterm of the sequence aann is the value of the term that areis the value of the term that arelooking forlooking for n is the number of the term you aren is the number of the term you aretrying to determinetrying to determine r is the common ratio between termsr is the common ratio between terms
Try this…Try this…Use the geometric rule to determine theUse the geometric rule to determine the1010ththterm of this sequence:term of this sequence:4, 20, 100, 5004, 20, 100, 500 aa11 = 4= 4 n = 10n = 10 r = 20/4 = 5r = 20/4 = 5aann == aa11**rr((nn-1)-1)== 44 ** 55((1010-1)-1)=7812500=7812500
Example of a GeometricExample of a GeometricSequence in the RealSequence in the RealWorldWorld Suppose you borrow $10,000Suppose you borrow $10,000from a bank that charges 5%from a bank that charges 5%interest. You want to determineinterest. You want to determinehow much you will owe the bankhow much you will owe the bankover a period of 5 years.over a period of 5 years.
SolutionSolution The first term in the sequences will beThe first term in the sequences will bethe initial amount of money borrowed,the initial amount of money borrowed,which is $10,000.which is $10,000. The common ratio is 105%, this can beThe common ratio is 105%, this can berepresented as 1.05 as a decimal.represented as 1.05 as a decimal. n is the number of years you have then is the number of years you have theloan.loan.aann = $10,000(1.05)= $10,000(1.05)(5-1)(5-1)
Assignment:Assignment:Complete practiceComplete practiceproblems fromproblems fromSection 12.2 & 12.3Section 12.2 & 12.3in your textbookin your textbook
Works CitedWorks CitedGeometric Sequences in the Real World.Geometric Sequences in the Real World. SophiaSophia. N.p., n.d. Web. 11 May 2013.. N.p., n.d. Web. 11 May 2013.McDougal Littell ClassZone.McDougal Littell ClassZone. ClassZoneClassZone. N.p., n.d. Web. 11 May 2013.. N.p., n.d. Web. 11 May 2013.There Is Video.Then, There Is Virtual Nerd.There Is Video.Then, There Is Virtual Nerd. How Do You Determine If a Sequence IsHow Do You Determine If a Sequence IsArithmetic or Geometric?Arithmetic or Geometric? N.p., n.d. Web. 11 May 2013.N.p., n.d. Web. 11 May 2013.