CHAPTER 12 SEQUENCES, PROBABILITY, AND STATISTICS

Transcription

1 CHAPTER SEQUENCES, PROBABILITY, AND STATISTICS

2 PRE-CALCULUS: A TEACHING TEXTBOOK Lesso 77 Sequeces We ve fiished with coic sectios, so ow we re goig to switch gears ad talk about sequeces. I simple laguage, a sequece is just a list of umbers that progress i a certai patter. Here are two examples., 4, 6, 8, 0,, 9, 7, 8, 4, O top, the umbers start at ad the icrease by each time. O bottom, the umbers are,, ad o up. So the expoet o the icreases by each time. But both sequeces show a patter. Oe big differece betwee the two, though, is that the top sequece is fiite; it stops with the umber. However, the bottom sequece is ifiite, sice the umbers just keep goig. That s what the dots mea. The last term,, just shows the formula that s used to geerate all of the umbers i the sequece. The first term, is. Next comes 9, which is, ad so o. The ca represet ay positive iteger. The umbers of a sequece are called terms, by the way. Ad the term is sometimes called the geeral term. Here are two more examples of sequeces, both of which are ifiite.,,, 4, 5, 0,,, 6, 0, 5,, 8, 6, 45, Sequeces as Fuctios A sequece ca qualify as a fuctio. Remember, the techical defiitio of a fuctio is that it ca have just oe output (y-value) for each iput (x-value). We ca thik of the terms of the sequece as the y-values. They re the rage. The the x-values are just the order i which the terms are listed. For example, here are the x- ad y-values for the sequece. Table 77. y-values x-values,9,7,8,4,...,,, 4, 5,... Sice o x-value has two matchig y-values, this sequece qualifies as a fuctio. The big differece betwee a sequece that s a fuctio ad most of the fuctios we ve bee workig with is that a sequece oly allows positive itegers i its domai. That s why we use the letter to stad for the idepedet variable. It s geerally uderstood that the letter icludes just positive itegers. (The letter k is also sometimes used.) Restrictig the domai to positive itegers limits the rage as well. The symbols for a sequece are a little differet from those for a fuctio. Istead of writig f ( ) =, we usually represet each term of the sequece by a letter with a little below to desigate the order i which the term appears. For example, the first term of is a ad the secod term is a. So we write a = = ad a = = 9. The geeral symbol for a term of the sequece is just a. Other letters besides a ca be used as well. 708

3 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS Sequeces ad Formulas It s possible to use a formula to represet a etire sequece. Istead of writig, 9, 7, 8, 4,, for istace, we could just write. That tells us the patter of the sequece, which is everythig we eed to kow. We ca use to fid ay specific term of the sequece we wat. To fid the 5 th term, we just put 5 i for like this. 5 a 5 = = 4 We ca fid ay other term the same way. This is called fidig the th term of a sequece. Aother example of a formula is 4. This represets the sequece below. 4 values, 7, 5, 9, -values,,, 4, 5,... To fid the 47 th term of 4, we just put 47 i for ad calculate the value of the formula. a 47 = 4(47) = 87 Thik of how much loger it would have take to fid this aswer by just listig the terms of the sequece oe-byoe, accordig to the patter. Most sequeces have patters where the ext term depeds o the value of the previous term. For istace, i the sequece, 7,, 5, 9, the ext term ca be calculated by just addig 4 to the previous term. The direct formula for this sequece is 4, as we showed. But we ca also write a formula for the sequece usig the previous term as the idepedet variable. If a represets ay particular term of the series, the a = a + 4. This formula ca also be used to geerate specific terms of the sequece. Startig with the term 9, we ca fid the very ext term like this. a = + a 4 = A formula that uses a previous term for its idepedet variable is called a recursive formula. Recursive formulas are usually a little easier to write (whe give several terms of the sequece), but they re ot as useful as direct formulas. With a recursive formula, for istace, we ca t fid the 47 th term of a sequece just by pluggig i 47 for the idepedet variable. Arithmetic Sequeces There are a couple of types of sequeces that are really importat because they re used so much i the real world. First, there s a arithmetic sequece, which is just a sequece where each term is obtaied by addig the same umber to the previous term. The sequeces, 4, 6, 8, 0, ad, 7,, 5, 9, are both arithmetic. The key feature of a arithmetic sequece is that the differece betwee ay two terms is always the same. This is called the commo differece. So the commo differece of the sequece, 4, 6, 8, 0, is ad the commo differece of the sequece, 7,, 5, 9 is 4., 4, 6, 8, 0, commo differece of, 7,, 5, 9 commo differece of 4 709

4 PRE-CALCULUS: A TEACHING TEXTBOOK More geerally, ay arithmetic sequece ca be writte i the form below, where a is the first term ad d is the commo differece. { a, a + d, a + d, a + d,..., a + ( ) d...} commo differece of d That meas i geeral the direct formula for ay arithmetic sequece is a = a + ( ) d. So if we re tryig to fid the th term of some arithmetic sequece that starts with the umber 9 ad has a commo differece of 4, we just put 9 i for a ad 4 i for d to get this. a = 9 + ( )(4) = 5 The geeral form of the recursive formula for a arithmetic sequece is writte like this. a = a + d (where ) If a is some term of the sequece, the previous term gives us a. That s all the recursive formula is sayig. a is the previous term. Addig the commo differece (d) to the The easiest way to tell whether a sequece is arithmetic is just to subtract several pairs of cosecutive terms ad see if you get the same aswer each time. If you do, the the sequece is arithmetic. If the differeces are ot all the same, the the sequece is t arithmetic. Geometric Sequeces The secod importat type of sequece is a geometric sequece. This is where the terms icrease by the same factor each time. The sequece is geometric, because to get each term we ca just multiply the previous term by., 9, 7, 8, 4,,, 9, 7, 8, Here s aother example. I this oe each term is times the value of the previous term., 4, 8, 6,,,, 4, 8, 6, The direct formula for this sequece is, ad the recursive formula is a = a (where ). So i a geometric sequece, istead of addig each previous term by the same umber, we multiply each previous term by the same umber. The umber we eed to multiply by is called the commo ratio. The commo ratio of, 4, 8, 6,, 64, is. The ame comes from the fact that the ratio of ay two cosecutive terms i the sequece is always equal to the commo ratio. For istace, dividig ay term i, 4, 8, 6,, 64, by the previous term always gives a aswer of. 4 =, 8 4 =, 6 8 =, 6 =, 64 = commo ratio = 70

5 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS This meas the easy way to recogize a geometric sequece is just to divide several pairs of cosecutive terms ad see if you get the same umber every time. There are lots of other geometric sequeces with differet startig poits ad differet costat ratios. Here are a few more., 6,, 4, 48, 96,, 4, 8, 6,,,, 9, 7, 8, 4, ( ) Notice that the top sequece has a costat i frot of the formula. That s ecessary i order to make the first term of the sequece equal to. After that each term is twice the size of the previous oe. If the first term of the sequece had bee 4, the the umber i frot of would have bee 4. The expoet is so that the first expoet will 0 equal 0. See, puttig i for gives this: = = =. The domai of a sequece is always the positive itegers, so you sometimes have to make adjustmets like this to start the sequece of with the proper first term. The sequece,,, 9, 7, 8, 4, ( ) is also iterestig. It has a egative commo ratio:. This causes the sig of the terms to alterate betwee positive ad egative. Ad otice that the i the expoet of the formula ( ) 0 causes the first term to equal, sice ( ) =. If the expoet were just, the the first term of the sequece would equal, because ( ) =. We ca use the formula of a geometric sequece to fid th term, just as we do with other sequeces. For istace, to fid the 7 th a 7 7 = =. 8 term of, we just put 7 i for ad calculate the value of the expressio: Geometric sequeces ca also be show with geeral symbols. The geeral for ay geometric sequece is writte like this (with a commo ratio of r). { a, ar, ar, ar,..., ar...} commo ratio = r The geeral form of a recursive formula for ay geometric sequece is a = ra, (where ). Arithmetic sequeces have the same form as liear fuctios. A easy way to see this is to compare the graphs of the two. Here are the graphs of the arithmetic sequece ad the liear fuctio y = x. Figure 77. y y=x (,4) arithmetic sequece (,) y (,4) (,) (-,-) (0,0) x x (-,-4) Oly itegers are i the domai 7

6 PRE-CALCULUS: A TEACHING TEXTBOOK The oly differece i the two fuctios is their domais. The sequece domai is limited to positive itegers, as always. That causes its graph to iclude oly a series of poits. However, the fuctio s domai icludes all real umbers, which makes its graph a smooth, cotiuous lie. The same kid of relatioship exists betwee a geometric sequece ad a expoetial fuctio. For istace, compare the graphs of the geometric sequece ad y = x. y y= x Figure 77. (4,8) geometric sequece y (4,8) (,) (,7) (,9) x (,) (,7) (,9) x Oly itegers are i the domai See, the sequece just has poits, but the fuctio is a smooth curve. That s because the sequece fuctio has a domai that s limited to positive itegers. Practice 77 a. Fid the ext three terms i the sequece : 9,,,... b. Write a direct formula ( th term) for the geometric sequece, 5, 5, 5, 65,? A. + B. 5 C. + 4 D. 5 E. 5 c. Fid the 7 th term i the geometric sequece with a = ad a commo ratio (r) of 4. d. Which of the followig equatios has roots of 4 ad? A. D. x 9x + 4 = 0 B. x + 7x 4 = 0 E. x + x 8x + 8 = 0 C. x + 9x + 4 = 0 x x = e. If 8si θ + siθ = 0, the what is the smallest positive value of θ i degrees? Estimate the aswer to oe decimal place. f. Solve the word problem below. The cost of a booth at a couty fair ca be represeted by the fuctio C( t) =,000 5t where t is the umber of years that the compay has reted a booth. Based o this model, what would be the price of a booth for a compay who has reted a booth for the past 6 years? 7

7 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS Problem Set 77 Tell whether each setece below is True or False.. A arithmetic sequece is a sequece where each term is obtaied by addig the same umber to the previous term.. A geometric sequece is a sequece where the terms are icreased by the same factor (multiplied by the same umber) each time. Aswer each questio below. y x. Select the ceter ad vertices of = 5 8 A. Ceter ( 5, 0 ) ; Vertices ( 8, 0), (8, 0) B. Ceter ( ) C. Ceter ( 0, 0 ) ; Vertices (0, 5), (0,5) D. Ceter ( ) E. Ceter ( 0, 0 ) ; Vertices ( 5, 0), (5,0) 0, 0 ; Vertices (0, 8), (0,8) 0,8 ; Vertices (0,5), (5,0) 4. Select the ceter ad vertices of ( x + 6) ( y 4) = 49 9 A. Ceter ( 6, 4) ; Vertices (, 0), (, 0) B. Ceter ( 6, 4) C. Ceter ( 0,0 ) ; Vertices (7,0), (0,) D. Ceter ( 6, 4) E. Ceter ( 6, 4) ; Vertices (, 4), (,4) ; Vertices (, 4), (, 4) ; Vertices ( 6,), ( 6,7) 5. If the x-axis is traslated 5 places dow ad the y-axis 4 places to the right, the equatio chages to which of the followig? y = x A. y = x 6 B. D. y = x E. y = x C. y = x y = x Tell whether each coic sectio below is a circle, parabola, ellipse or hyperbola without completig the square? 6. x + y x + y = x x y y = 0 Fid the ext three terms i each sequece below. 8. 4, 8,, 6 9., 5, 75, 75 (a) 0. 4,,,... 7

8 PRE-CALCULUS: A TEACHING TEXTBOOK Select the correct aswer for each sequece below.. Write a direct formula ( th term) for the arithmetic sequece 7, 4,, 8,? A. 7 7 B. 7 C. D. + 7 E. 7 7 (b). Write a direct formula ( th term) for the geometric sequece, 4, 6, 64, 56,? A. 4 B. 4 C. + 4 D. 4 E. +. Write a recursive formula for the arithmetic sequece, 5, 8,,? A. a = B. a = a + C. a = a D. a = a + E. a = 4. Write a recursive formula for the arithmetic sequece,,, 9,? A. a = a B. a = D. a = a 5 E. a = a C. a = Aswer each questio below. 5. Fid the 7 th term i the arithmetic sequece with a = 5 ad a commo differece (d) of. (c) 6. Fid the 6 th term i the geometric sequece with a = ad a commo ratio (r) of. 7. Fid the commo ratio (r) i the geometric sequece:,,, Select the aswer for each questio below. 8. If f ( x) = x ad f ( g( x)) = x, the g( x ) =? A. D. x B. x C. 9x x E. x (d) 9. Which of the followig equatios has roots of ad? A. D. x + 5x = 0 B. x 7x + = 0 E. x + 7x + = 0 C. x 5x = 0 x + 5x 4x = 0 74

9 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS 0. If f ( k) = 5 k, the f =? 4 A. 5 B. 5 C. D. 5 E Aswer the questio below. Estimate the aswer to oe decimal place. (e). If 6si α siα 0 + =, the what is the smallest positive value of α i degrees? Solve the word problem below. (f). The aual cost of belogig to a certai coutry club ca be represeted by the fuctio C( t) = 7,000 00t where t is the umber of years a member has beloged to the coutry club. Based o this model, how much would the aual cost be for a member who had beloged for 5 years? 75

10 PRE-CALCULUS: A TEACHING TEXTBOOK Lesso 78 The Sum of a Sequece I the last lesso, we started learig about sequeces, which are just lists of umbers that progress i a patter. Sometimes it s useful to be able to calculate the sum of the terms i a sequece. Here s a simple example., 4, 6, 8, 0,, 4,, Let s fid the sum of the first seve terms of this sequece. All we have to do is add those terms together to get = 56 Sigma Notatio There s a short way to write the sum of a sequece. We use the capital Greek letter sigma, which is writte. This is the Greek capital S, so sigma stads for sum. To show the sum of the sequece above, we write = below. That says we re startig with the first term of the sequece. The we write 7 above, to say that we re edig with the 7 th term of the sequece. The expressio 7 = 7 = = 56 meas Start with the first term of the sequece ad add all the terms up through the 7 th term. Ad we already saw that that sum equals 56. Here are several other examples of showig sums of sequeces usig sigma otatio. 8 = = 04 = 4 4 = = 0 = 5 k = 7 = = k Formulas for Sums Sice arithmetic sequeces are so commo, mathematicias have worked out formulas for calculatig their sums. If { a, a, a,..., a } is some fiite arithmetic sequece, the the sum of its terms ca be calculated with the formula below. ( a + a ) sum of arithmetic sequece This looks kid of complicated, but the expressio ( a + a ) is just sayig to add the first ad last term of the sequece whose sum is beig calculated. The multiply that total by the umber of terms divided by. To see if it works, let s try the formula o a sequece that starts with 5 ad has a commo differece is. The first 6 terms of this sequece have a sum of 60. 5, 7, 9,,, = 60 6 Let s see if we get the right aswer with the formula: ( a + a ) equals (5 + 5), which comes out to 60. It works. 76

11 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS There s a well-kow story about this formula. Whe Friedrich Gauss, oe of the greatest mathematical geiuses of all time, was just 0 years old, his teacher assiged everyoe i his school class the task of addig all of the whole umbers from to 00 (just to keep everyoe busy). After just a few secods, Gauss wrote dow the aswer ad haded it to the teacher. Assumig that Gauss had just guessed, the teacher did t eve bother lookig at the aswer, ad made the boy sit back dow ad cotiue workig. Later that day, the teacher happeed to glace at the aswer ad was astoished to fid the umber 5,050, which is exactly right. Gauss had realized that the umbers through 00 ca be paired up so that every pair has a sum of 0. All you have to do is take the first umber plus the last umber (+ 00 ) ad work your way iward, two at a time. Every pair has a sum of Sice the umber of pairs is half of 00, the sum of through 00 has to be 00 0 or 5,050. Sice is the same as 00 ad a + a equals 0, this is just the formula ( a + a ) applied to the specific sequece of the first 00 positive itegers. Gauss had figured all of this out i his head i just a few secods (without ayoe ever teachig him about sequeces)! There s also a formula for calculatig the sum of ay fiite geometric sequece. If { a, a, a,..., a } is a fiite geometric sequece, the the sum ca be calculated with the formula below. ( a ) r r sum of geometric sequece Remember, r is the commo ratio of the sequece ad is the umber of terms that are beig added. Let s use this formula to calculate the sum of the first 5 terms of the geometric sequece. Here are the first 5 terms., 6,, 4, 48 The terms add to 9. Applyig the formula, we get the followig. It works. A couple of other examples are show below. ( ) ( 5 a ) ( ) r = = = 9 r sum of first 4 terms of = ( ) ( 4 a ) ( 5) r = = = 0 r sum of first 5 terms of ( ) 5 4 a ( r ) ( ) ( ) = = = r 4 = The story may or may ot be true. 77

12 PRE-CALCULUS: A TEACHING TEXTBOOK Sum of a Ifiite Sequece Whe calculatig the sum of a sequece, it s commo sese that the sequece would have to be fiite. After all, how could you add up a ifiite umber of terms? The aswer would obviously be ifiite. For istace, look how fast the sums of the geometric sequece icrease as we iclude more ad more terms = = =, =, = 9, = 9, 5 The sum of this sequece is obviously headed toward ifiity. By the way, whe we add just some of the terms of a sequece, the aswer is called a partial sum. So 0, 6,,09 ad the other aswers above are partial sums of the sequece. The poit is that the sum of a ifiite sequece does t seem to have commo sese. But mathematics is t based o commo sese; it s based o logic ad proof. Ad believe-it-or-ot, there are some ifiite sequeces that actually ca be added up to get a umerical aswer. Here s a example of oe , This is the sum of the ifiite geometric sequece,. Now look at several of the partial sums. + + = or = or = or = or = or = or

13 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS If we were to cotiue this process, addig more ad more terms, the partial sums would get closer ad closer to 6, but ever quite reach that umber. I fact, sice this is a geometric sequece we ca calculate some partial sums with larger umbers of terms. Remember, the formula ( a r ) ca be used to calculate the sum of a fiite r geometric sequece. I this case, a = ad r =. That gives us this. ( ) Now let s calculate the partial sum whe = 50. ( ) ( ) 50 = Notice how small is the umber 50. That makes the quatity iside the paretheses just a tiy bit less tha. ( ) Multiplyig by gives a product o top that s ever so slightly less tha The last step is to ivert the ad multiply. ( )() The aswer is just a little uder 6. Are you startig to get the idea here? No matter how may terms of the ifiite sequece we add, the sum is always uder 6. Ad the more terms we add, the closer the sum gets to 6. It ever quite reaches 6 exactly, though. The techical word for this kid of situatio is limit. We say that as approaches ifiity, the expressio approaches a limit of 6. It s basically the same cocept as a fuctio approachig a asymptote. The graph gets closer ad closer to some lie, but ever crosses it. We ll talk more about limits at the ed of this course. Limits are of the very greatest importace i calculus. For ow, the mai poit is that sice the partial sums of the sequece get closer ad closer to 6, mathematicias decided to defie the sum of the ifiite sequece (where keeps goig to ifiity) to equal 6 exactly. That s how it s possible for some ifiite sequeces to have sums. It all depeds o whether their partial sums, as the terms icrease, approach some limit. 79

14 PRE-CALCULUS: A TEACHING TEXTBOOK There are some other techical words related to this topic that you should kow. A expressio cotaiig a ifiite umber of terms added together is called a ifiite series. So the ifiite sequece of qualifies as a ifiite series. I geeral form, we write a ifiite series like this. = a = a + a + a a +... Notice that the ifiity symbol is placed o top of sigma to show that goes all the way to ifiity. A ifiite series that approaches a limit (ad actually has a sum) is said to coverge. A ifiite series, such as the oe for, that does ot approach a limit is said to diverge. There are tests that ca be applied to determie whether a particular series coverges or diverges, but they usually require more advaced math kowledge. There s oe covergece test, however, that you ca lear right ow. It s the test for ay geometric series. As it turs out, a geometric series will always coverge if its commo ratio (r) is betwee ad. If it s outside that rage, the the series will diverge (ad ot have a sum). Covergece test for geometric series < commo ratio < The series for coverged to 6, because the commo ratio was, which is betwee ad. However, 4 the geometric series does ot coverge, because it s commo ratio is. That s outside the rage 4 of < r <. As we saw above, the partial sums of grow faster ad faster, without approachig ay limit. This covergece test for ay geometric series makes a lot of sese. The test is really sayig that the commo ratio has to be a proper fractio i order for the series to approach a limit. Whe the commo ratio is a proper fractio, each term will be smaller tha the previous oe by that fractio. If the commo ratio is, the each term will be oe-half of the previous term. If the commo ratio is, each term will be oe-third of the previous term. Whe the commo ratio is a proper fractio, the terms get smaller ad smaller. So it makes sese that the sum could approach a limit. Whe the commo ratio is greater tha or less tha, the magitude of each term will be bigger tha the previous oe. Practice 78 a. Write a direct formula ( th term) for the arithmetic sequece 5,,, 9,? A. 8 B. 8 C. 8 D. 8 8 E. 8 b. Fid the sum of the first 70 terms of the arithmetic sequece The tests for covergece are very complex. It s ot eough for each term to be smaller tha the previous oe. Some series with smaller ad smaller terms still do t coverge. A kowledge of calculus is required to determie whether most series actually coverge. 70

15 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS c. Use the covergece test to determie whether the geometric sum + ( ) or diverges? coverges d. Name a regio of the xy plae whose poits satisfy either the iequality y < x or the iequality 0 y <. A. a square B. the half of the plae below the x-axis C. a circle D. the regio of the plae bouded by a parabola E. a crescet-shaped regio i the plae e. Solve the word problem below. The divided paid o a stock have icreased at rate of % sice 996. The value of the divided ca be represeted by the fuctio ( ) 500(.0) t C t = where t is time i years after 996. Based o this, what is the total amout oe would have eared from the divideds i 004 if oe had held the stock sice 997? Estimate your aswer to the earest cets. Problem Set 78 Tell whether each setece below is True or False.. To calculate the sum of ay arithmetic sequece, add the first ad last term of the sequece the multiply that total by the umber of terms divided by.. The formula for calculatig the sum of ay fiite geometric sequece is ( a + a ). For each pair of rectagular coordiates below, select the correct coversio ito polar coordiates. Do t use the polar coversio fuctio o your calculator.. G (,48) A. (5.4, 66.4 ) B. ( 5.4,.6 ) C. ( 5.4, 66.4 ) D. (69, 66.4 ) E. (5.4, 66.4 ) 4. H ( 5,75) A. ( 9., 4.7 ) B. (9., 4.7 ) C. (9.,4.7 ) D. ( 9.,4.7 ) E. (, 55. ) Select the stadard form for each coic sectio equatio below. 5. y y x = 0 A. D. x x + = ( y ) B. + = ( y ) E. x + = ( y ) C. x = ( y ) y = ( x ) 7

16 PRE-CALCULUS: A TEACHING TEXTBOOK 6. 9x 54x + 6y + 64y = A. D. ( x ) ( y + ) + = B. 4 6( x ) 9( y ) = E. y 6 = 9( x ) C. ( x ) ( y + ) + = 4 x y + = 6 9 Fid the ext three terms i each sequece below , 7, 4, 8. 4, 8, 6, Select the correct aswer for each sequece below. (a) 9. Write a direct formula ( th term) for the arithmetic sequece 8,, 8,,? A. 5 B. 5 C. 5 + D. 5 5 E Write a recursive formula for the geometric sequece 4,,, A. a = a 6 B. a = D. a = a E. a = a,? C. a = Aswer each questio below.. Fid the 4 th term i the arithmetic sequece with a = ad a commo differece (d) of 6.. Fid the commo ratio (r) i the geometric sequece: 6, 4, 94,, Fid the commo differece (d) for the arithmetic sequece with a = 5 ad a = Fid the sum of the whole umbers from to 50. (b) 5. Fid the sum of the first 60 terms of the arithmetic sequece Use the covergece test to determie whether each geometric sum below coverges or diverges? (c) ( )

17 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS Select the aswer for each questio below. (d) 8. Name a regio of the xy plae whose poits satisfy either the iequality y > x or the iequality 0 y >. A. a circle B. a crescet-shaped regio i the plae C. the half of the plae above the x-axis D. the regio of the plae bouded by a parabola E. a square 9. Some umber is added to the three umbers,, 7 ad 67, to create the first three terms of a geometric sequece. What is the value of? A. 0 B. C. D. 4 E. 5 Aswer the questio below. 0. If f x ( ) 7x 5 =, the f ( f ()) =? Estimate your aswer to a whole umber. A lie has parametric equatios x = t 9 ad y = 5t, give t is the parameter. What is the y-itercept of the lie that represets the direct relatioship betwee x ad y? Solve the word problem below. (f). Aual isurace premiums have icreased at a rate of 0% sice 990. The aual cost of isurace ca be represeted by the fuctio ( ),000(.) t C t = where t is time i years after 990. Based o this, what is the total amout oe would have paid by 00 to the isurace compay if oe had bee a member sice 99? Estimate your aswer to the earest cets. 7

18 PRE-CALCULUS: A TEACHING TEXTBOOK Lesso 79 More o Ifiite Series I the last lesso, we leared that some ifiite series have a fiite sum, which is really surprisig. Those series are said to coverge, ad the others that do t have a fiite sum diverge. Ad remember, ay geometric series will coverge if its commo ratio is betwee ad. It s ice to be able to tell whether a particular geometric series coverges. But it s icer still to be able to calculate quickly the actual sum of a covergig series. I the last lesso, we did that with the geometric series for. We used the formula ( ) a r ( ). After pluggig i the values for, the formula became. r ( ) Remember, as icreased, got closer ad closer to 6. Formula for a Ifiite Geometric Series We could have come up with 6 a lot easier by just aalyzig the expressio ifiity, the term ( ). As approaches gets smaller ad smaller. To be more accurate, it approaches a limit of 0. That meas, to ( ) calculate the limit of as approached ifiity, all we have to do is replace Whe approaches ifiity ( ) ( 0) with 0. That gives us this. Now we ca simplify ormally to get the limit of 6. = 6 The same kid of aalysis ca be performed o ay other geometric series with a commo ratio betwee ad. We ca get the limits for those as well. Fortuately, there s a eve easier way to calculate limits for covergig geometric series. We ca take the limit of the geeral formula ( a r ). Whe < r <, we kow that r will approach 0 as x approaches ifiity. r So we ca just replace r with 0 ad simplify. a ( 0) r a r a The formula ca be used to immediately calculate the limit of ay covergig geometric series. The series r below has a iitial term of 5 ad a commo ratio of. So a = 5 ad r =

19 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS Pluggig a = 5 ad r = ito the formula gives us a limit of 7.5. a r That meas if we were to keep addig the terms of the series forever, the partial sums would a get closer ad closer to 7.5 but ever quite reach it. Whe usig the formula, just remember that it oly works r for geometric series (ot other kids). Ad it oly works whe the commo ratio is betwee ad. It s kid of iterestig that a repeatig decimal is really just a ifiite geometric series. Take 0., for example. This meas , which is geometric, ad has a commo ratio of. Ad, of 0 00, 000 0, course, the fractios just keep goig forever. Whe studets are first taught that is the same as 0., they re too youg to appreciate how strage it is that a ifiite umber of fractios could add up to equal the fiite sum. They re just taught to memorize that ad 0. are the same. But ow you kow that some ifiite series do a have fiite sums. Whe such a series is geometric, the sum ca be calculated with the formula. Let s prove r that does actually equal 0 00, 000 0, 000. The first term is 0. So we eed to put that i for a. Ad the commo ratio is 0, so that should go i for r. a r 0 0 Simplifyig gives us this It works. All repeatig decimals are really ifiite geometric series. Ad sice our umber system is based o 0s, a they all have a commo ratio of. So we ca use the formula (with r = ) to covert ay repeatig 0 r 0 decimal ito a fractio. 75

20 PRE-CALCULUS: A TEACHING TEXTBOOK Mathematical Iductio There s a method of doig mathematical proofs that s particularly useful whe workig with sequeces of positive itegers. The method is called mathematical iductio. To show you how it works, let s say that we wat to prove that the sum of odd positive itegers always equals = We could try to prove this by checkig for several values of. For istace, if =, the rule works. The rule also works for = = 9 = = 5 5 = 5 The problem with checkig for specific values is that it does t prove that works for all possible values for. That s much tougher to prove, because could be aythig, eve some absolutely huge umber. As you may remember from geometry, showig that a rule is true for several idividual cases does t qualify as a mathematical (deductive) proof. But we ca do a proper proof usig mathematical iductio. The first step is to show that the iitial term of the sequece for =, i other words. Not surprisigly, it works. = = is right for just Next, we assume that is right for some other value for. We ll call it k. Usig the expressio, the k th term of the sequece must be k. That gives us the equatio below (k ) = k Assumig that this equatio is true, let s ow put i the ext term o the left. The term followig (k ) ca be obtaied by substitutig k + for k i the expressio. That gives us ( k + ). Of course, if we add a quatity to the left side of a equatio, we have to add the same quatity to the right side k + k + = k + k ( ) ( ) ( ) Next, we simplify the right side k + k + = k + k ( ) ( ) The is just the formula for geeratig ay particular term of this sequece. Puttig i for makes equal, which is the first odd iteger. Puttig i for makes equal, which is the secod odd iteger, ad so o. 76

21 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS The equatio ow says that the series through the k + th term is equal to k + k +. If that s really true, the we should get k + k + whe we substitute k + for i the expressio. That gives us ( k + ), which simplifies to equal k + k + k + or k + k +, so it works. That fiishes the proof. Now let s thik about what we just did. We proved that our rule works for =, which is the very first term of the sequece. The we proved that if the rule works for some value = k, the it must also work for = k +, which is the sequece with the very ext term added. That word if is importat, because we just assumed that the rule worked for k, remember. But what if k =? We kow the rule works for. But we ve already proved that it also must work for the ext value for which is. But the we ca let k =. Ad we ve proved that if the rule works for, the it also must work for. Sice we ve proved that the rule works for the first term () ad that it will always work for the very ext value for, the we have effectively prove that the rule works for ay all the way up to ifiity. That s how mathematical iductio ca be used to prove thigs that are true for all positive itegers. Practice 79 a. If the commo ratio (r) is ad a 6 = 6, fid the first term of the geometric sequece. b. Fid the sum of the ifiite series your reasoig , or state that the sum does ot exist ad explai c. Covert the repeatig decimal ito a fractio. d. Use mathematical iductio to prove that the propositio positive itegral values of = is valid for all e. If the 9 th term of a arithmetic sequece is 95, ad the 7 th term is 75, the what is the first term of the sequece? f. Solve the word problem below. Durig a physical exercise drill called a suicide, athletes have to ru to the first marked lie the back to the startig poit. They the tur aroud ad ru to the ext marked lie ad the back agai to the startig poit. This repeats util the athlete reaches the fial marked lie. The distace i feet ra for each lie ca be represeted by the fuctio D( l) = (5 l) where l is the umber of the lie the athlete ra to ad from. How may feet would a athlete have ru total if he had just fiished ruig to ad from the 6 th lie? Problem Set 79 Tell whether each setece below is True or False.. It is impossible to calculate the sum of a ifiite series.. Mathematical iductio is a method for doig proofs that s particularly useful whe workig with sequeces of positive itegers. 77

22 PRE-CALCULUS: A TEACHING TEXTBOOK Select the equatio for each coic sectio below.. The ceter of a ellipse is at (, 5), the legth of the major axis is 4 uits, ad the legth of the mior axis is 6 uits. A. D. ( x + 5) ( y ) + = 49 9 B. ( x ) ( y + 5) + = 7 E. x y + = C ( x ) ( y + 5) = 7 ( x ) ( y + 5) + = 8 4. Select the ceter ad vertices of the hyperbola y x = A. Ceter ( 0, 0 ) ; Vertices (0, 7), (0, 7) B. Ceter ( ) C. Ceter ( 8, 0 ) ; Vertices ( 7, 0), (7, 0) D. Ceter ( ) E. Ceter ( 0, 0 ) ; Vertices (0, 8), (0,8) 0,7 ; Vertices (0,8), (8,0) 0, 0 ; Vertices ( 8, 0), (8,0) Select the correct aswer for each sequece below. 5. Write a recursive formula for the arithmetic sequece 5.5, 7, 8.5, 0,? A. a.5 = B. a = a +.5 C. a = a 4 D. a =.5 E. a =.5a 6. Write a direct formula ( th term) for the geometric sequece 4, 8, 7, 9? A. D. 4 B. E. C. 4 Aswer each questio below. 7. Fid the 0 th term i the arithmetic sequece with a = ad a commo differece (d) of. 8. Fid the commo ratio (r) i the followig geometric sequece:,, 8,, 8... (a) 9. If the commo ratio (r) of a geometric sequece is ad a 5 =, fid the first term of the sequece. 0. Fid the sum of the first 40 terms of the arithmetic sequece

23 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS Fid the sum of each ifiite geometric series, or idicate that the sum does ot exist (b) Covert each repeatig decimal below ito a fractio (c) Use mathematical iductio to prove that the equatio below is true for all positive iteger values of. (d) 5. ( + ) = Select the aswer for each questio below. 6. Poit ( 6, 4) is located o a circle i the coordiate plae give by the equatio Which of the followig poits must also lie o the circle? A. (, ) B. (, ) C. (,.5) D. ( 7,4) E. (., ) ( x ) ( y ) r + + =. 7. I a arithmetic sequece, a6 = a 0 ad a 4 =. Betwee which two cosecutive terms does 0 lie? A. a 4 ad a 5 B. a 5 ad a 6 C. a 6 ad a 7 D. a 7 ad a 8 E. a 9 ad a 0 8. I a geometric series, suppose g = 4 ad g 5 = 08. What is g =? g A. B. C. D. 4 E. 8 Aswer the questio below. 9. What is the remaider whe the polyomial 4 x x x is divided by x +? 0. What is the maximum value of f x = x ( ) 4 ( 6)? (e). If the 7 th term of a arithmetic sequece is 80, ad the 9 th term is 00, the what is the first term of the sequece? Solve the word problem below. (f). Durig a physical exercise drill called a suicide, athletes have to ru to the first marked lie the back to the startig poit. They the tur aroud ad ru to the ext marked lie ad the back agai to the startig poit. This repeats util the athlete reaches the fial marked lie. The distace i feet ra for each lie ca be represeted by the fuctio D( l) = (0 l) where l is the umber of the lie the athlete ra to ad from. How may feet would a athlete have ru total if he had just fiished ruig to ad from the 0 th lie? 79

24 PRE-CALCULUS: A TEACHING TEXTBOOK Lesso 80 Probability The word probability is used a lot i day-to-day coversatio. People will say that the probability of rai is 60% or the probability of a cadidate wiig a electio is 75%. Basically, probability is a measure of how certai somethig is to happe i the future is. Probabilities ca be writte as percetages, fractios, or decimals. So the probabilities 60% ad 75% ca also be writte as 0.60 ad 0.75 or as 5 ad 4 (which are the fractios ad 75 fully reduced). Whe a evet is absolutely certai to happe, the probability is 00% or. Whe a evet will 00 defiitely ot happe, the probability is 0% or just 0. So all probabilities rage from 0 to. Calculatig Probability How is probability actually calculated? Well, the first step is to cout all of the possible ways that somethig ca happe. That s the total umber of possible outcomes. As a example, imagie flippig a coi. There are two possibilities. The coi ca lad heads or tails. The total umber of possible outcomes is therefore equal to. That umber goes i the bottom of a fractio. Now let s say we wat to calculate the probability that the coi will lad heads. Heads is the favorable outcome. Sice there s just oe way the coi ca lad heads, the umber of favorable outcomes equals, which goes i the top of the fractio. Number of favorable outcomes = Total umber of outcomes The fractio equals, which meas the probability of the coi ladig heads is or 0.5 or 50%, depedig o how you wat to write it. That s how to calculate probability. It s the umber of favorable outcomes (what you re calculatig the probability of) divided by the total umber of outcomes. Table 80. Calculatig probability Number of favorable outcomes Probability= Total umber of outcomes This does t mea that a coi is guarateed to lad heads oe time out of every two flips. Probabilities are t that certai. A probability of meas that if you flip a coi may, may times, the umber of heads will be approximately equal to of all the flips. Also, the method of calculatig probabilities (dividig favorable outcomes by total outcomes) oly works whe the possibilities are all equally likely. What if a Olympic swimmer was cosiderig swimmig i the shallow ed of a pool? There are two possibilities, he could drow or ot drow. Does that mea that he has a or 50% probability of drowig? No, of course ot. The aalysis is wrog, because the two possibilities are t equally likely. A expert swimmer is extremely ulikely to drow i the shallow ed of a pool. The probability might be. Sice a coi is symmetrical i shape, we assume that the coi is just as 00,000 likely to lad heads as tails. That s why we ca calculate the probability by dividig favorable outcomes by total possible outcomes. 70

25 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS As aother example, let s calculate the probability of rollig a eve umber with a sigle die. There are 6 possible outcomes:,,, 4, 5, ad 6. So 6 goes i the bottom of the fractio. The favorable outcomes are the eve umbers, which are, 4, ad 6. That s favorable outcomes. So we eed to put i the top of the fractio. Number of favorable outcomes = = Total umber of outcomes 6 That gives us 6 or, which meas that the probability of rollig a eve umber with a die is also equal to. Sometimes we eed to calculate the probability of two evets happeig. For istace, we might wat to calculate the probability of flippig a coi heads two times i a row. I cases like this, figurig out the total umber of outcomes is a little harder. Oe way to do it is to draw a tree diagram. O the first flip, there are two possibilities, so we draw two braches of the tree. If the coi lads heads, the we will flip it a secod time, ad there will be two possibilities agai. So we draw two braches extedig out from the first heads. If the coi lads tails o the first flip, the we will also flip a secod time. So we draw two more braches extedig out from the first tails. Figure 80. Heads Heads First toss Tails Tails Heads Tails Secod toss To fid the total umber of possible outcomes, we just have to cout all the braches o the far right of the diagram. There are 4 braches, which meas flippig a coi twice has 4 possible outcomes. We eed to put 4 i the bottom of the fractio. To get the favorable outcomes, we just eed to fid how may braches ivolve two heads. There s just oe. The brach o top. So goes o top of the fractio. Number of favorable outcomes = Total umber of outcomes 4 The probability of flippig a coi twice ad havig it lad heads both times is or 5%. 4 7

26 PRE-CALCULUS: A TEACHING TEXTBOOK A tree diagram is eve more helpful i calculatig the probability of rollig a 0 with two dice. There are 6 possibilities whe rollig the first die. The die could lad,,, 4, 5, or 6. So that s 6 braches to start. Whichever oe of the cases occurs, we will roll the secod die. Ad there will be 6 possibilities for it. So we eed to draw 6 braches extedig from each of the 6 origial possibilities. Figure 80. First roll 4 5 Secod roll To get the total umber of possible outcomes, we just cout all the possibilities o the far right. There are 6. So 6 goes i the bottom of the fractio. How do we fid the favorable outcomes? We just cout the umber of outcomes o the right where the dice add to 0. There s 4 ad 6, 5, ad 5, 6, ad 4. That s favorable outcomes, so goes i the top of the fractio. Number of favorable outcomes = = Total umber of outcomes 6 After reducig, we ed up with a probability of or about 8.%. That s pretty low. Ve diagrams, which you may remember from geometry, ca also be helpful whe calculatig probabilities. Ve diagrams are just circles (or other shapes) that represet a group or set of objects (or elemets). Puttig oe circle iside the other meas that the ier circle is a subset of the big oe. For example, people livig i Califoria are a subset of people livig i the U.S. Figure 80. Uited States Califoria 7

27 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS If two sets have o members i commo, the the circles are etirely separate. The set of wome ad the set of me are a example. Figure 80.4 Me Wome Sometimes, i figurig out a probability, we might eed to iclude all the objects (elemets) i a set A as well as all of the objects (elemets) i aother set B. This is called the uio of A ad B ad it s writte like this: A B. Figure 80.5 A B The uio of A ad B equals A+B Whe two sets partially overlap, the sets are said to itersect. The itersectio icludes all the objects that are members of both sets. The itersectio of two sets A ad B is draw by overlappig the circles (see below). The symbol is A B. Figure 80.6 A B Itersectio The itersectio of A ad B equals the objects i both sets Here s how Ve diagrams might be used to calculate a probability. What if we wated to calculate the probability of drawig a diamod or a face card from a deck of cards. Just i case you re ot that familiar with cards, a card deck has 5 cards i total. There are 4 types of cards: diamods, hearts, clubs, ad spades. 7

28 PRE-CALCULUS: A TEACHING TEXTBOOK Each type represets of the deck, which is cards each. A face card is a card that has a face o it. There are 4 three types of face cards: jacks, quees, ad kigs. There are face cards i each of the 4 categories of cards (diamods, hearts, clubs, ad spades). The first step i calculatig the probability of drawig a diamod or a face card is to figure out the total possible outcomes. That s just 5, sice there are 5 cards i total. We ll draw a rectagle to represet the etire deck of cards. The diamods are a subset of the etire deck ad so are all the face cards. Importatly, though, the set of diamods itersects with the set of face cards. That s because of the face cards are also diamods. Figure 80.7 Deck Diamods 0 Face cards 9 5 To figure out the total favorable outcomes, we eed to add the umber of diamods to the umber of face cards ad the subtract the itersectio betwee the two. The subtractio prevets us from double-coutig the diamod face cards (that are i both sets). There are diamods, face cards ( for each of the 4 categories of cards). The itersectio betwee the two sets is. So we get + =. The umber of favorable outcomes, the, is. The probability of drawig a diamod or a face card is 5 or, which is about 4.%. 6 Number of favorable outcomes = = Total umber of outcomes 5 6 Multiplicatio Priciple of Coutig It ca be pretty tedious to draw tree diagrams every time you wat to cout the total outcomes or the favorable outcomes i a situatio. That s why there are coutig shortcuts that ca be used istead. Whe a probability calculatio ivolves multiple evets (like two coi flips or two dice), the rule is that the total umber of outcomes is equal to the product of the possible outcomes of each evet. If the total umber of outcomes of evet A is symbolized as (A) ad the total umber of outcomes of evet B is symbolized as (B), the the followig is true. Multiplicatio Priciple of Coutig Table 80. (A ad the B)=(A). (B) 74

29 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS Usig the multiplicatio priciple of coutig o flippig a coi twice, we just take the possibilities of the first flip times the possibilities of the secod flip to get a total umber of outcomes of 4, which we kow is right. st flip d flip outcomes x outcomes = 4 total outcomes Similarly, we ca use the shortcut o rollig dice. The first roll has 6 possible outcomes ad the secod roll also has 6 possible outcomes. Sice 6 times 6 equals 6, there must be 6 possible outcomes for both rolls. This is also correct. st roll d roll 6 outcomes x 6 outcomes = 6 total outcomes The multiplicatio priciple of coutig ca be applied o matter how may evets are ivolved. Here s a tougher example. Sam wats to password his persoal webpage. If he uses pass-words with 6 characters, letters followed by umbers, how may possible passwords ca be created? To do this problem, the first step is to imagie 6 blak spaces where the letters ad umbers go. Three letters Three umbers Sice the alphabet has 6 letters, there are 6 possibilities that could go i the first letter blak, 6 i the secod, ad 6 i the third. Sice our umbers have 0 possible digits, there are 0 possibilities that could go i the first umber blak, 0 i the secod, ad 0 i the third. Usig the multiplicatio priciple of coutig, we get the followig = 7, 576, 000 Believe-it-or-ot there are 7,576,000 differet letter- umber passwords that ca be created. I a problem like this, drawig a tree diagram to cout all the outcomes would be impossible. Additio Priciple of Coutig There s also a additio priciple of coutig. This arises whe we eed to iclude objects from two differet sets. For istace, we might eed to calculate the probability that the people i a group are either from Califoria or Texas. I this case, the favorable outcomes would iclude the uio of those two sets, which requires the sets to be added. Here s the rule stated formally. Table 80. Additio Priciple of Coutig (A or B)=(A)+(B) 75

30 PRE-CALCULUS: A TEACHING TEXTBOOK Of course if the sets itersected, the we have to subtract the umber i the itersectio set. That might be the case if some people had homes i both Califoria ad Texas. Table 80.4 Additio Priciple of Coutig with Itersectio (A or B)=(A)+(B)-(A B) This is the same priciple we used to calculate the probability of drawig a diamod or a face card from a deck. Oly istead of drawig a Ve diagram, we could have just used the above formula. Practice 80 a. Covert the repeatig decimal ito a fractio. b. Use mathematical iductio to prove that the equatio positive iteger values of. ( ) ( ) = is true for all c. There are 8 black, 4 avy, blue ad 6 white pairs of socks i a drawer. Without lookig, Jim pulled out oe pair. What is the probability that he chose a black or avy pair of socks? d. I a group of 5 studets, 8 studets ca play the piao, studets ca play the guitar, ad 4 studets ca play both piao ad guitar. If oe studet is chose at radom from this group to participate i a musical evet, what is the probability that this studet ca play the piao or the guitar? e. A home security compay offers a security system that uses a security code. How may differet security codes ca be created if each code has 4 letters followed by umbers? Problem Set 80 Tell whether each setece below is True or False.. Whe a evet is absolutely certai to happe, the probability is 00% (or ), ad whe a evet will defiitely ot happe, the probability is 0% (or 0).. To calculate probability, take the umber of favorable outcomes divided by the total umber of outcomes. Aswer each questio below.. Fid the 9 th term i the arithmetic sequece with a = 8 ad a commo differece (d) of 4. If the commo ratio (r) of a geometric sequece is ad g 7 =, 458, fid the first term of the sequece. 5. Fid the sum of the whole umbers from to

31 CHAPTER : SEQUENCES, PROBABILITY, AND STATISTICS Fid the sum of each ifiite geometric series, or idicate that the sum does ot exist Covert each repeatig decimal below ito a fractio (a) Use mathematical iductio to prove that the equatio below is true for all positive iteger values of. (b) (4 ) = ( ) Aswer each questio below.. A box cotais 5 red, blue, ad white marbles. Oe marble is chose at radom. What is the probability that the chose marble is white? What is the probability that the chose marble is blue?. If a die is rolled two times, what is the probability of rollig fours? Write your aswer as a fractio.. What is the probability of rollig a 9 with two dice? Write your aswer as a fractio. (c) 4. There are bottles of grape juice, 6 bottles of orage juice, 9 bottles of apple juice, ad bottles of fruit puch i the cooler. Without lookig, Matthew chose a bottle for himself. What is the probability that he chose a bottle of orage juice or apple juice? (d) 5. I a group of 0 studets, 5 studets play o a basketball team, 9 studets play o a soccer team, ad studets play o both a basketball ad soccer team. If oe studet is chose at radom from this group, what is the probability that he or she is o a basketball or soccer team? Write your aswer as a fractio. (e) 6. How may differet automobile licese plates ca be created if each plate has letters followed by umbers? Select the aswer for each questio below. y x 7. Which of the followig is a y-itercept of the hyperbola =? 5 64 A. (0, 5) B. (5,0) C. (0,8) D. (0, 5) E. (0,65) 8. If the patter of the terms, 8, 6 cotiues, which of the followig would be the 8 th term of the sequece? A. 8 B. D. 7 ( ) E. 8 ( ) C. 9 8 ( ) 77