Chapter 11: Sequences and Series

1 DiscreteMathematicsDiscreteMathematics57 4 Unit 4 Discrete MathematicsDiscrete mathematicsis the branch ofmathematics thatinvolves finite ordiscontinuousquantities. In thisunit, you will learnabout Sequences , Series , probability,and KayeProfessor of MathematicsUniversity of BirminghamChapter 11 Sequences and SeriesChapter 12 Probability and StatisticsChapter 11 Sequences and SeriesChapter 12 Probability and StatisticsSource: USA TODAY, November 3, 2000 Minesweeper, a seemingly simple game included on most personal computers, could help mathematicianscrack one of the field s most intriguing problems. The buzz began after Richard Kaye, a mathematicsprofessor at the University of Birmingham in England,started playing Minesweeper.

2 After playing the game steadily for a few weeks, Kaye realized thatMinesweeper, if played on a much larger grid, has thesame mathematical characteristics as other problemsdeemed insolvable. In this project, you will research a mathematician of the past and his or her role in thedevelopment of discrete continue workingon your WebQuest asyou study Unit on to your WebQuest by reading the Task. Minesweeper : Secret to Age-Old Puzzle?Unit 4 Discrete Mathematics57511-712-1616635 LessonPage Source: Census Bureau Current Population Report (Sept. 99)93%80%70%63%32%25%GamesWord processingInternetE-mailEducationalprogr amSchoolassignmentsWhy teens use PCs at homeThe leading purposes teens age 12 to 17 gave for usinga PC at home:By Mark Pearson and Jerry Mosemak, USA TODAYUSA TODAY Snapshots 576 Chapter 11 Sequences and SeriesSequences andSeries arithmetic sequence (p.)

3 578) arithmetic Series (p. 583) sigma notation (p. 585) geometric sequence (p. 588) geometric Series (p. 594)Key VocabularyMany number patterns found in nature and used in business can bemodeled by Sequences , which are lists of numbers. Some sequencesare classified by the method used to predict the next term from theprevious term(s). When the terms of a sequence are added, a Series Lesson 11-2, you will learn how the number of seats in the rows of an amphitheater can be modeled using a Series . Lessons 11-1 through 11-5 Use arithmetic andgeometric Sequences and Series . Lesson 11-6 Use special Sequences and iteratefunctions. Lesson 11-7 Expand powers by using theBinomial Theorem.

4 Lesson 11-8 Prove statements by usingmathematical Chapter 11 Sequences and SeriesChapter 11 Sequences and Series577 Sequences and SeriesMake this Foldable to help you organize your with one sheet of 11" by 17" paper and four sheets of notebook and WritingAs you read and study the Chapter , fill the journal with examples for each SkillsTo be successful in this Chapter , you ll need to masterthese skills and be able to apply them in problem-solving situations. Reviewthese skills before beginning Chapter Lessons 11-1 and 11-3 Solve EquationsSolve each equation.(For review, see Lessons 1-3 and 5-5.) 12 4x2. 40 10 3x 2x45.

5 18 4 20 For Lessons 11-1 and 11-5 Graph FunctionsGraph each function.(For review, see Lesson 2-1.)7.{(1, 1), (2, 3), (3, 5), (4, 7), (5, 9)}8.{(1, 20), (2, 16), (3, 12), (4, 8), (5, 4)}9. (1, 64), (2, 16), (3, 4), (4, 1), 5, 14 10. (1, 2), (2, 3), 3, 72 , 4, 145 , 5, 381 For Lessons 11-1 through 11-5, 11-8 Evaluate ExpressionsEvaluate each expression for the given value(s) of the variable(s).(For review, see Lesson 1-1.) (y 1)zif x 3, y 8, and z 212. x2 (y z) if x 10, y 3, and z bc 1if a 2, b= 12 , and c 714. a(11 bbc)2 if a 2, b 3, and c 515. 1 ab if a 12 , and b 16 16. n(n2 1) if n 10 Fold and CutStaple and LabelFold the short sidesof the 11'' by 17''paper to meet in the notebookpaper in halflengthwise.

6 Inserttwo sheets ofnotebook paper ineach tab and staplethe edges. Labelwith SEQUENCESThe numbers 3, 4, 5,6, .., representing the number of shingles in each row, are an example of a sequence of numbers. Aisa list of numbers in a particular order. Each number in asequence is called a . The first term is symbolizedby a1, the second term is symbolized by a2, and so graph represents the information from the tableabove. A sequence is a discrete function whose domainis the set of positive integers. Many Sequences have patterns. For example, in thesequence above for the number of shingles, each termcan be found by adding 1 to the previous term.

7 Asequence of this type is called an arithmetic sequence . An is a sequence in which each term after the first is found by adding a constant, called the d, to the previous differencearithmetic sequencetermsequenceyx456721034268101214 16182022 ShinglesRowVocabulary sequence term arithmetic sequence common difference arithmetic meansArithmetic Sequences578 Chapter 11 Sequences and Series Use arithmetic Sequences . Find arithmetic is nailing shingles to the roof of a house in overlapping rows. There are three shingles in the top row. Since the roof widens from top to bottom, one additional shingle is needed in each successive the Next Terms Find the next four terms of the arithmetic sequence 55, 49, 43.

8 Find the common difference dby subtracting two consecutive 55 6 and 43 49 6So, d add 6 to the third term of the sequence , and then continue adding 6 untilthe next four terms are ( 6) ( 6) ( 6) ( 6)The next four terms of the sequence are 37, 31, 25, and numbers in asequence may not beordered. For example, thenumbers 33, 25, 36, 40,36, 66, 63, 50, .. are asequence that representsthe number of home runsSammy Sosa hit in eachyear beginning with TipIt is possible to develop a formula for each term of an arithmetic sequence interms of the first term a1and the common difference d. Consider Example Shinglesare arithmetic Sequences related to roofing?

9 Are arithmetic Sequences related to roofing?Lesson 11-1 Arithmetic Sequences579 The following formula generalizes this pattern for any arithmetic an Equation for the nth TermWrite an equation for the nth term of the arithmetic sequence 8, 17, 26, 35, ..In this sequence , a1 8 and d 9. Use the nth term formula to write an a1 (n 1)dFormula for nth terman 8 (n 1)9a1= 8, d= 9an 8 9n 9 Distributive Propertyan 9n equation is an 9n Term of an Arithmetic SequenceThe nth term anof an arithmetic sequence with first term a1and commondifference dis given byan a1 (n 1)d,where nis any positive The table below showstypical costs for aconstruction company torent a crane for one, two,three, or four : 0( 6)55 1( 6)55 2( 6)55 3( 6).

10 55 (n 1)( 6)a1 0 .da1 1 .da1 2 .da1 3 . (n 1)dSequencenumberssymbolsExpressed in numbersTerms ofdand symbolsthe First TermMonthsCost ($)175,000290,0003105,0004120,000 Find a Particular Term CONSTRUCTIONR efer to the information at the left. Assuming that thearithmetic sequence continues, how much would it cost to rent the crane fortwelve months?ExploreSince the difference between any two successive costs is $15,000, thecosts form an arithmetic sequence with common difference 15, can use the formula for the nth term of an arithmetic sequencewith a1 75,000 and d 15,000 to find a12, the cost for twelve a1 (n 1)dFormula for nth terma12 75,000 (12 1)15,000n 12, a1 75,000, d 15,000a12 240, would cost $240,000 to rent the crane for twelve can find terms of the sequence by adding 15,000.