Algebra II – Sequences and Series | Mathematics

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Study GuideAlgebra II–Sequences and Series1.Definition and Examples of SequencesAsequenceis anordered list of numbersthat follows a certain pattern.Examples of sequences include:Thethree dots (…)show that the pattern continues.Each number in a sequence is called aterm.For example, in the sequence:1, 3, 5, 7, 9, …•1 is thefirst term•3 is thesecond term•5 is thethird termTerms are usually written using notation like:Here:•(a1) =first term•(a2) = second term•(an) =nth termThe expression for (an) is called thegeneral termornth-term formula. It allows us to findany termin the sequence.

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Study GuideExample 1:Finding Terms from a FormulaSuppose the general term is:We want to find thefirst five terms.First termSecond termThird termFourth termFifth termResultThe first five terms are:5, 8, 11, 14, 17Example 2:Another Sequence FormulaNow consider the formula:Find the first five terms.First term

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Study GuideSecond termThird termFourth termFifth termResultThe first five terms are:2, 6, 18, 54, 162Example 3:Finding the Formula for a SequenceSometimes we are given a sequence and must find theformula for the nth term.Sequence 12, 4, 6, 8, …Notice that each term istwice the term number.Formula

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Study GuideSequence 210, 50, 250, 1250, …Each term is5 times the previous term, so this is ageometric sequence.Rewrite the numbers:FormulaSequence 33, 7, 11, 15, 19, …The pattern increases by4 each time, so it is anarithmetic sequence.Write each term in terms of (n):Formula

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Study GuideSimplify:Key Ideas to Remember•Asequenceis an ordered list of numbers that follows a pattern.•Each number in the sequence is called aterm.•Thenth-term formula (an)allows us to find any term in the sequence.•Some sequences increase by a constant amount (arithmetic sequences).•Others multiply by a constant number (geometric sequences).SummarySequences describe number patterns that follow specific rules.•The notation (a1, a2, a3,…) represents the terms of a sequence.•The formula (an) gives thegeneral rulefor generating the sequence.•Arithmetic sequences grow byadding a constant difference.•Geometric sequences grow bymultiplying by a constant ratio.2.Quiz: Definition and Examples of Sequences1. QuestionFind thenext termin the sequence:

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Study GuideAnswer Choices•6•7•8Correct Answer:7Why This Is CorrectLook at the difference between terms:So the sequenceincreases by 8 each time.Next term:So the next term is7.2. QuestionFind the7th termof the sequence described by:Answer Choices•27•31•35Correct Answer:27

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Study GuideWhy This Is CorrectSubstitute (n = 7):So the7th term is 27.3. QuestionFind the9th termof the sequence described by:Answer Choices•64•32•16Correct Answer:16Why This Is CorrectSubstitute (n = 9):So the9th term is 16.4. QuestionFind the5th termof the sequence described by:

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Study GuideAnswer Choices•91•213•253Correct Answer:253Why This Is CorrectSubstitute (n = 5):So the5th term is 253.5. QuestionFind the11th termof the sequence described by:Answer Choices•130•154•180Correct Answer:154Why This Is Correct

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Study GuideSubstitute (n = 11):So the11th term is 154.3.Arithmetic SequenceAnarithmetic sequenceis a sequence of numbers where each term after the first is found byadding the same number to the previous term.This constant number is called thecommon difference.General FormulaThe formula for the (n)thterm of an arithmetic sequence is:Where:•(an) = the (n)th term•(a1) = the first term•(d) = common difference•(n) = term numberWhat is the Common Difference?Thecommon differenceis the amount added to each term to get the next term.It can be found using:This means you subtract the first term from the second term.

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Study GuideExample 1Find thecommon difference, write the formula for the sequence, and determine the20th term.Sequence 11, 5, 9, 13, 17, …Step 1: Find the CommonDifferenceSo the common difference is4.Step 2: Write the FormulaUsing the formulaSubstitute the values:Simplify:Step 3: Find the 20th TermSo the20th term is 77.

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