Ch 9 Sequences, Series, and Probability

Transcription

1 Ch 9 Sequeces, Series, ad Probability Have you ever bee to a casio ad played blackjack? It is the oly game i the casio that you ca wi based o the Law of large umbers. I the early 1990s a group of math ad sciece majors from the Massachusetts Istitute of Techology (MIT) devised a foolproof scheme to wi at blackjack. A professor at MIT developed a basic strategy outlied i the figure that is based o the probability of combiatios of particular cards beig dealt, give certai cards already showig. To play blackjack (also called 21), each perso is dealt two cards with the optio of takig additioal cards. The goal is to get a combiatio of cards that is worth 21 poits (or less) without goig over (called a bust). You have to avoid goig over 21 or stayig too far below 21. All face cards (jacks, quees, ad kigs) are worth 10 poits, ad a ace i blackjack is worth either 1 or 11 poits. The studets used the professor s strategy alog with a card-coutig techique to place high bets whe there were more high-value cards left i the deck. It is reported that i 1992 the team wo $4,000,000 from Las Vegas casios. The casios caught o ad the studets were all baed withi 2 years. The 2008 movie 21 was based o this evet. 1

2 IN THIS CHAPTER we will discuss coutig ad probability i additio to three other topics: sequeces ad mathematical iductio, ad the biomial theorem. SECTION 9.1 SEQUENCES AND SERIES 9.2 ARITHMETIC SEQUENCES AND SERIES 9.3 GEOMETRIC SEQUENCES AND SERIES 9.4 MATHEMATICAL INDUCTION 9.5 THE BINOMIAL THEOREM 9.6 COUNTING, PERMUTATIONS, ANDCOMBINATIONS 9.7 PROBABILITY CHAPTER OBJECTIVES Uderstad the differece betwee sequeces ad series. Fid the geeral, th, term of a sequece or series. Uderstad the differece betwee fiite ad ifiite series. Evaluate a fiite series. Determie if a ifiite series coverges or diverges. Prove a mathematical statemet usig iductio. Use the biomial theorem to expad a biomial raised to a positive iteger power. Uderstad the differece betwee permutatios ad combiatios. Calculate the probability of a evet. Page 705 2

3 Page 706 SECTION 9.1 SEQUENCES AND SERIES SKILLS OBJECTIVES Fid terms of a sequece give the geeral term. Look for a patter i a sequece ad fid the geeral term Apply factorial otatio. Apply recursio formulas. Use summatio (sigma) otatio to represet a series. Evaluate a series. CONCEPTUAL OBJECTIVES Uderstad the differece betwee a sequece ad a series. Uderstad the differece betwee a fiite series ad a ifiite series. Sequeces The word sequece meas a order i which oe thig follows aother i successio I mathematics, it meas the same thig. For example, if we write x, 2 x, 3 x, 4 x, 5 x,?, what would the ext term i the sequece be, the oe where the questio mark ow stads? The aswer is 6 6x. DEFINITION Sequece A sequece is a fuctio whose domai is a set of positive itegers. The fuctio values, or terms of the sequece are writte as a1, a2 a3 a,,,, Rather tha usig fuctio otatio, sequeces are usually writte with subscript (or idex) otatio, a subscript. A fiite sequece has the domai { 1, 2, 3,..., } for some positive iteger. A ifiite sequece has the domai of all positive itegers { 1, 2, 3,...}. There are times whe it is coveiet to start the idexig at 0 istead of 1: 3

4 ,,,,, a0 a1, a2 a3 a Sometimes a patter i the sequece ca be obtaied ad the sequece ca be writte usig a geeral term. I the previous example, ad coefficiet. We ca write this sequece as called the geeral term x, 2 x, 3 x, 4 x, 5 x,6x,, each term has the same expoet a = x, = , 6 where a is EXAMPLE 1 Fidig the Sequece, Give the Geeral Term Fid the first four ( = 1, 2, 3, 4) terms of the sequeces, give the geeral term. Page 706 Page 707 4

5 Fid the first four terms of the sequece a = ( ) 1 2 EXAMPLE 2 Fidig the Geeral Term, Give Several Terms of the Sequece Fid the geeral term of the sequece, give the first five terms. 5

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7 Fid the geeral term of the sequece, give the first five terms. Parts (b) i both Example 1 ad Example 2 are called alteratig sequeces, because the terms alterate sigs (positive ad egative). If the odd terms, a1, a3, a 5,..., are egative ad the eve terms, a2, a4, a 6,..., are positive, we iclude ( 1) i the geeral term. If the opposite is true, ad the odd terms are positive ad the eve terms are egative, we iclude ( 1) + 1 i the geeral term. Factorial Notatio May importat sequeces that arise i mathematics ivolve terms that are defied with products of cosecutive positive itegers. The products are expressed i factorial otatio. DEFINITION Factorial If is a positive iteger, the! (stated as factorial ) is the product of all positive itegers from dow to 1. ( )( )! = ad 0! = 1 ad 1! = 1 7

8 The values of! for the first six oegative itegers are 0! = 1 1! = 1 2! = 2 1 = 2 3! = 3 2 l = 6 4! = = 24 5! = = 120 Notice that 4! = =4 3!. I geeral, we ca apply the formula = ( ) Ofte the brackets are ot used, ad the otatio ( )! 1!.! = 1! implies calculatig the factorial ( - 1)! ad the multiplyig that quatity by. For example, to fid 6!, we employ the relatioship! = ( - 1)! ad set = 6: 6! = 6 5! = = 720 8

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10 EXAMPLE 3 Fidig the Terms of a Sequece Ivolvig Factorials Fid the first four terms of the sequece, give the geeral term a = x! EXAMPLE 4 Evaluatig Expressios with Factorials Evaluate each factorial expressio. 10

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13 COMMON MISTAKE YOUR TURN Evaluate the factorial expressios. 13

14 Recursio Formulas Aother way to defie a sequece is recursively, or usig a recursio formula. The first few terms are listed, ad the recursio formula determies the remaiig terms based o previous terms. For example, the famous Fiboacci sequece is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,. Each term i the Fiboacci sequece is foud by addig the previous two terms. We ca defie the Fiboacci sequece usig a geeral term: a = 1, a = 1, ad a = a + a The Fiboacci sequece is foud i places we least expect them (for example, pieapples, broccoli, ad flowers). The umber of petals i a flower is a Fiboacci umber. For example, a wild rose has 5 petals, lilies ad irises have 3 petals, ad daisies have 34, 55, or eve 89 petals. The umber of spirals i art Italia broccoli is a Fiboacci umber (13). 14

15 EXAMPLE 5 Usig a Recursio Formula to Fid a Sequece YOUR T U RN Fid the first four terms of the sequece: Sums ad Series Whe we add the terms i a sequece, the result is a series. DEFINITION Series Give the ifiite sequece a1, a2, a3,, a, sequece is called a ifiite series ad is deoted by a + a + a + + a the sum of all of the terms i the ifiite ad the sum of oly the first terms is called a fiite series, or th partial sum, ad is deoted by S = a + a + a + + a

16 The capital Greek letter Σ (sigma) correspods to the capital S i our alphabet. Therefore, we use Σ as a shorthad way to represet a sum (series). For example, the sum of the first five terms of the sequece 1, 4, 9, 16, 25,., 2,.. ca be represeted usig sigma (or summatio) otatio: This is read the sum as goes from 1 to 5 of 2. The letter is called the idex of summatio, ad ofte other letters are used istead of. It is importat to ote that the sum1 ca start at other umbers besides 1. If we wated the sum of all of the terms i the sequece, we would represet that ifiite series usig summatio otatio as 16

17 EXAMPLE 6 Writig a Series Usig Sigma Notatio Write the followig series usig sigma otatio. 17

18 YOUR TURN Write the followig series usig sigma otatio. Now that we are comfortable with sigma (summatio) otatio, let s tur our attetio to evaluatig a series (calculatig the sum). You ca always evaluate a fiite series. However, you caot always evaluate a ifiite series. 18

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20 EXAMPLE 7 Evaluatig a Fiite Series Study Tip The sum of a fiite series always - exists. The sum of a ifiite series may or may ot exist. Ifiite series may or may ot have a fiite sum. For example, if we keep addig , the there is o sigle real umber that the series sums to because the sum cotiues to grow without boud. However, if we add ± this sum is = 0.9, which is a ratioal umber, ad it ca be prove that 0.9 = 1. 20

21 EXAMPLE 8 Evaluatig a Ifiite Series, If Possible Evaluate the followig ifiite series, if possible. 21

22 Solutio (b): Expad the series. = This sum is ifiite sice it cotiues to grow without ay boud. I part (a) we say that the series coverges to ad i part (b) the say that the series diverges. YOUR TURN Evaluate the followig ifiite series, if possible. 22

23 Applicatios The aual sales at Home Depot from 2000 to 2002 ca be approximated by the model 2 a = , where a, is the yearly sales i billios of dollars = 0,1, 2. What does the fiite series a = 0 tell us? It tells us the average yearly sales over 3 yes 23

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25 I Exercises 1-12, write the first four terms of the sequece. Assume starts at a = 2. a = 2 3. a = a = x 5. a = ( + 1) 25

26 6. a = ( + 1) 7. a = 2! 8. a =! 1! ( + ) ( ) a = x + 26

27 + ( ) a = a = ( 1) ( + 1)( + 2) 12. a = ( 1) ( + 1)

28 I Exercises 13-20, fid the idicated term of the sequece a = a9 =? 2 ( ) ( + ) 1! 15. a = a19 =? 2! 28

29 l 20. a = e a49 =? 29

30 I Exercises , write a expressio for the th term of the give sequece. 30

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32 I Exercises 29-40, simplify the ratio of factorials. Solutios:

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34 I Exercises 41-50, write the first four terms of the sequece defied by the recursio formula. Assume the sequece begis at 1. 34

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36 I Exercises 51-64, evaluate the fiite series. 36

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39 I Exercises 65-68, evaluate the ifiite series, if possible. I Exercises 69-76, apply sigma otatio to write the sum. 39

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42 77. Moey. Upo graduatio Jessica receives a commissio from the U.S. Navy to become a officer ad a $20,000 sigig bous for selectig aviatio. She puts the etire bous i a accout that ears 6% iterest compouded mothly. The balace i the accout after moths is Her commitmet to the Navy is 6 years. Calculate A 72. What does A 72 represet? 78. Moey. Dyla sells his car i his freshma year ad puts $7,000 i a accout that ears 5% iterest compouded quarterly. The balace i the accout after quarters is Calculate A 12. What does A 12, represet? 79. Salary. A attorey is tryig to calculate the costs associated with goig ito private practice. If she hires a paralegal to assist her, she will have to pay the paralegal $20.00 per hour. To be competitive with most firms, she will have to give her paralegal a $2 per hour raise per year. Fid a geeral term of a sequece a, which represets the hourly salary of a paralegal with years of experiece. What will be the paralegal s salary with 20 years of experiece? 42

43 80. TL Salaries. A player i the NFL typically has a career that lasts 3 years. The practice squad makes the league miimum of $275,000 (2004) i the first year, with a $75,000 raise per year. Write the geeral term of a sequece a,, that represets the salary of a NFL player makig the league miimum durig his etire career. Assumig = 1 correspods to the first year, what does 3 a represet? = 1 43