9.1. Sequences and Series. Sequences. What you should learn. Why you should learn it. Definition of Sequence

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1 _9.qxd // : AM Page Chapter 9 Sequeces, Series, ad Probability 9. Sequeces ad Series What you should lear Use sequece otatio to write the terms of sequeces. Use factorial otatio. Use summatio otatio to write sums. Fid the sums of ifiite series. Use sequeces ad series to model ad solve real-life problems. Why you should lear it Sequeces ad series ca be used to model real-life problems. For istace, i Exercise 9 o page, sequeces are used to model the umber of Best Buy stores from 99 through. Sequeces I mathematics, the word sequece is used i much the same way as i ordiary Eglish. Sayig that a collectio is listed i sequece meas that it is ordered so that it has a first member, a secod member, a third member, ad so o. Mathematically, you ca thik of a sequece as a fuctio whose domai is the set of positive itegers. f a, f a, f a, f a,..., f a,... Rather tha usig fuctio otatio, however, sequeces are usually writte usig subscript otatio, as idicated i the followig defiitio. Defiitio of Sequece A ifiite sequece is a fuctio whose domai is the set of positive itegers. The fuctio values a, a, a, a,..., a,... are the terms of the sequece. If the domai of the fuctio cosists of the first positive itegers oly, the sequece is a fiite sequece. O occasio it is coveiet to begi subscriptig a sequece with istead of so that the terms of the sequece become a, a, a, a,.... Example Writig the Terms of a Sequece Scott Olso /Getty Images The HM mathspace CD-ROM ad Eduspace for this text cotai additioal resources related to the cocepts discussed i this chapter. Write the first four terms of the sequeces give by a. a b. a. Solutio a. The first four terms of the sequece give by a are a a a 7 a. st term d term rd term th term b. The first four terms of the sequece give by a are a a a a. st term d term rd term th term Now try Exercise.

2 _9.qxd // : AM Page Sectio 9. Sequeces ad Series Exploratio Write out the first five terms of the sequece whose th term is a. Are they the same as the first five terms of the sequece i Example? If ot, how do they differ? Additioal Example Write a expressio for the apparet th term a of the sequece...,,,, Solutio: :... Terms:... a Apparet patter: Each term has a umerator that is greater tha its deomiator, which implies that a. To graph a sequece usig a graphig utility, set the mode to sequece ad dot ad eter the sequece. The graph of the sequece i Example (a) is show below. You ca use the trace feature or value feature to idetify the terms. Techology Example A Sequece Whose Terms Alterate i Sig Write the first five terms of the sequece give by Solutio The first five terms of the sequece are as follows. a a a a a 7 9 Now try Exercise 7. st term d term rd term th term th term Simply listig the first few terms is ot sufficiet to defie a uique sequece the th term must be give. To see this, cosider the followig sequeces, both of which have the same first three terms.,,,,...,,...,,,,...,,... Example Fidig the th Term of a Sequece Write a expressio for the apparet th term a of each sequece. a.,,, 7,... b.,,, 7,... Solutio a. :... Terms: 7... a Apparet patter: Each term is less tha twice a., which implies that b. :... Terms: 7... a Apparet patter: The terms have alteratig sigs with those i the eve positios beig egative. Each term is more tha the square of, which implies that a Now try Exercise 7. a.

3 _9.qxd // : AM Page Chapter 9 Sequeces, Series, ad Probability Some sequeces are defied recursively. To defie a sequece recursively, you eed to be give oe or more of the first few terms. All other terms of the sequece are the defied usig previous terms. A well-kow example is the Fiboacci sequece show i Example. Example The Fiboacci Sequece: A Recursive Sequece The subscripts of a sequece make up the domai of the sequece ad they serve to idetify the locatio of a term withi the sequece. For example, a is the fourth term of the sequece, ad a is the th term of the sequece. Ay variable ca be used as a subscript. The most commoly used variable subscripts i sequece ad series otatio are i, j, k, ad. The Fiboacci sequece is defied recursively, as follows. a, a, a k a k a k, where k Write the first six terms of this sequece. Solutio a th term is give. a st term is give. a a a a a Use recursio formula. a a a a a Use recursio formula. a a a a a Use recursio formula. a a a a a Use recursio formula. Now try Exercise. Factorial Notatio Some very importat sequeces i mathematics ivolve terms that are defied with special types of products called factorials. Defiitio of Factorial If is a positive iteger, factorial is defied as!.... As a special case, zero factorial is defied as!. Here are some values of! for the first several oegative itegers. Notice that! is by defiitio.!!!!!! The value of does ot have to be very large before the value of! becomes extremely large. For istace,!,,.

4 _9.qxd // : AM Page Factorials follow the same covetios for order of operatios as do expoets. For istace, whereas!!...!.... Sectio 9. Sequeces ad Series Example Writig the Terms of a Sequece Ivolvig Factorials Write the first five terms of the sequece give by a!. Begi with. The graph the terms o a set of coordiate axes. Solutio a! th term a! st term a FIGURE 9. Additioal Examples! a.!! b.!! c.! You may wat to poit out to your studets that!! Note i Example (a) that you ca simplify the computatio as follows.! 7!!!!! 7 d term rd term th term Figure 9. shows the first five terms of the sequece. Now try Exercise 9. Whe workig with fractios ivolvig factorials, you will ofte fid that the fractios ca be reduced to simplify the computatios. Evaluatig Factorial Expressios Evaluate each factorial expressio.!!! a. b. c.!!!! Solutio a. b. c. a! a! a! Example! 7 7!!!!!!!! Now try Exercise 9.!!

5 _9.qxd // : AM Page Chapter 9 Sequeces, Series, ad Probability Techology Most graphig utilities are able to sum the first terms of a sequece. Check your user s guide for a sum sequece feature or a series feature. Readig ad writig the upper ad lower limits of summatio correctly will help with problems ivolvig upper ad lower limits i calculus. Summatio Notatio There is a coveiet otatio for the sum of the terms of a fiite sequece. It is called summatio otatio or sigma otatio because it ivolves the use of the uppercase Greek letter sigma, writte as. Defiitio of Summatio Notatio The sum of the first terms of a sequece is represeted by a i a a a a... a where i is called the idex of summatio, is the upper limit of summatio, ad is the lower limit of summatio. Example 7 Summatio Notatio for Sums Summatio otatio is a istructio to add the terms of a sequece. From the defiitio at the right, the upper limit of summatio tells you where to ed the sum. Summatio otatio helps you geerate the appropriate terms of the sequece prior to fidig the actual sum, which may be uclear. Fid each sum. a. i b. k c. Solutio a. b. i k k k i i! c. i i!!!!!!!! 7!! 7,.7 For this summatio, ote that the sum is very close to the irratioal umber e.7. It ca be show that as more terms of the sequece whose th term is! are added, the sum becomes closer ad closer to e. Now try Exercise 7. I Example 7, ote that the lower limit of a summatio does ot have to be. Also ote that the idex of summatio does ot have to be the letter i. For istace, i part (b), the letter k is the idex of summatio.

6 _9.qxd // : AM Page 7 Sectio 9. Sequeces ad Series 7 Variatios i the upper ad lower limits of summatio ca produce quite differet-lookig summatio otatios for the same sum. For example, the followig two sums have the same terms. i i Properties of Sums. c c, c is a costat.. a i, c is a costat... a i b i a i a i b i a i For proofs of these properties, see Proofs i Mathematics o page 7. Series b i May applicatios ivolve the sum of the terms of a fiite or ifiite sequece. Such a sum is called a series. Defiitio of Series ca i c b i Cosider the ifiite sequece a, a, a,..., a i,.... The sum of the first terms of the sequece is called a fiite series or the th partial sum of the sequece ad is deoted by a a a... a a i.. The sum of all the terms of the ifiite sequece is called a ifiite series ad is deoted by a a a... a i... a i. Example Fidig the Sum of a Series For the series Solutio a. The third partial sum is b. The sum of the series is i, i fid (a) the third partial sum ad (b) the sum i Now try Exercise

7 _9.qxd // : AM Page Chapter 9 Sequeces, Series, ad Probability Applicatio Sequeces have may applicatios i busiess ad sciece. Oe such applicatio is illustrated i Example 9. Example 9 Populatio of the Uited States Activities. Write the first five terms of the sequece. (Assume that begis with.) a 7 9 Aswer:,,,,. Write a expressio for the apparet th term of the sequece,,.,, Aswer:!. Fid the sum. k k k Aswer: For the years 9 to, the residet populatio of the Uited States ca be approximated by the model a.9..,,,..., where a is the populatio (i millios) ad represets the year, with correspodig to 9. Fid the last five terms of this fiite sequece, which represet the U.S. populatio for the years 999 to. (Source: U.S. Cesus Bureau) Solutio The last five terms of this fiite sequece are as follows. a populatio a populatio a.9... populatio a populatio a populatio Now try Exercise. Exploratio A cube is created usig 7 uit cubes (a uit cube has a legth, width, ad height of uit) ad oly the faces of each cube that are visible are paited blue (see Figure 9.). Complete the table below to determie how may uit cubes of the cube have blue faces, blue face, blue faces, ad blue faces. Do the same for a cube, a cube, ad a cube ad add your results to the table below. What type of patter do you observe i the table? Write a formula you could use to determie the colum values for a cube. Number of blue cube faces FIGURE 9.

8 _9.qxd // : AM Page 9 Sectio 9. Sequeces ad Series 9 9. Exercises The HM mathspace CD-ROM ad Eduspace for this text cotai step-by-step solutios to all odd-umbered exercises. They also provide Tutorial Exercises for additioal help. VOCABULARY CHECK: Fill i the blaks.. A is a fuctio whose domai is the set of positive itegers.. The fuctio values a, a, a, a,... are called the of a sequece.. A sequece is a sequece if the domai of the fuctio cosists of the first positive itegers.. If you are give oe or more of the first few terms of a sequece, ad all other terms of the sequece are defied usig previous terms, the the sequece is said to be defied.. If is a positive iteger, is defied as!..... The otatio used to represet the sum of the terms of a fiite sequece is or sigma otatio. 7. For the sum a i, i is called the of summatio, is the limit of summatio, ad is the limit of summatio.. The sum of the terms of a fiite or ifiite sequece is called a. 9. The of a sequece is the sum of the first terms of the sequece. PREREQUISITE SKILLS REVIEW: Practice ad review algebra skills eeded for this sectio at a I Exercises, write the first five terms of the sequece. (Assume that begis with.). a.. a. a. a. a 7. a. a 9.. a a. a. a. a.. a. 7. a. a 9. a. a.. a. a I Exercises, fid the idicated term of the sequece.. a. a a. a. a a a a a a I Exercises 7, use a graphig utility to graph the first terms of the sequece. (Assume that begis with.) 7. a. a 9. a. a.7.. a. a I Exercises, match the sequece with the graph of its first terms. [The graphs are labeled (a), (b), (c), ad (d).] (a) a (b) a (c) a (d). a. a. a. a.! a

9 _9.qxd // : AM Page Chapter 9 Sequeces, Series, ad Probability I Exercises 7, write a expressio for the apparet th term of the sequece. (Assume that begis with.). 7.,, 7,,,...., 7,,, 9,... 9.,,,,,....,,,,,....,,,,,,, 7,...,....., 9, 7,,,, 7, 9,...,....,, 9,,,....,,,,,... 7.,,,,, ,,,,,....,, 7,,,... I Exercises, write the first five terms of the sequece defied recursively.. a,. a, a k a k a k a k.. a, a, a k a k a k a k I Exercises, write the first five terms of the sequece defied recursively. Use the patter to write the th term of the sequece as a fuctio of. (Assume that begis with.). a,. a, a k a k a k a k 7.. a, a, a k a k a k a k,,,,,,... I Exercises 9, write the first five terms of the sequece. (Assume that begis with.) 9. a. a!!. a. a!!. a. a!! I Exercises 7, simplify the factorial expressio..!!.!! 7.!!.!! 9.!! 7.!! 7.!! 7.!! I Exercises 7, fid the sum. 7. i i i i i 79.. k k j j. k k. i i k.. i j j I Exercises, use a calculator to fid the sum.. j. 7.. I Exercises 9 9, use sigma otatio to write the sum k j k k k k k k! j... 9 i k j I Exercises 99, fid the idicated partial sum of the series. 99. i. i Fourth partial sum Fifth partial sum.. Third partial sum Fourth partial sum

10 _9.qxd // : AM Page Sectio 9. Sequeces ad Series I Exercises, fid the sum of the ifiite series..... i k k 7 k k i 7. Compoud Iterest A deposit of $ is made i a accout that ears % iterest compouded quarterly. The balace i the accout after quarters is give by A., (a) Write the first eight terms of this sequece. (b) Fid the balace i this accout after years by fidig the th term of the sequece.. Compoud Iterest A deposit of $ is made each moth i a accout that ears % iterest compouded mothly. The balace i the accout after moths is give by A.,,,,....,,,.... (a) Write the first six terms of this sequece. (b) Fid the balace i this accout after years by fidig the th term of the sequece. (c) Fid the balace i this accout after years by fidig the th term of the sequece. Model It 9. Data Aalysis: Number of Stores The table shows the umbers a of Best Buy stores for the years 99 to. (Source: Best Buy Compay, Ic.) Year Number of stores, a Model It (cotiued) (a) Use the regressio feature of a graphig utility to fid a liear sequece that models the data. Let represet the year, with correspodig to 99. (b) Use the regressio feature of a graphig utility to fid a quadratic sequece that models the data. (c) Evaluate the sequeces from parts (a) ad (b) for, 9,...,. Compare these values with those show i the table. Which model is a better fit for the data? Explai. (d) Which model do you thik would better predict the umber of Best Buy stores i the future? Use the model you chose to predict the umber of Best Buy stores i.. Medicie The umbers (i thousads) of AIDS cases reported from 99 to ca be approximated by the model a ,,,..., where is the year, with correspodig to 99. (Source: U.S. Ceters for Disease Cotrol ad Prevetio) (a) Fid the terms of this fiite sequece. Use the statistical plottig feature of a graphig utility to costruct a bar graph that represets the sequece. (b) What does the graph i part (a) say about reported cases of AIDS?. Federal Debt From 99 to, the federal debt of the Uited States rose from just over $ trillio to almost $7 trillio. The federal debt a (i billios of dollars) from 99 to is approximated by the model a ,,,..., a where is the year, with correspodig to 99. (Source: U.S. Office of Maagemet ad Budget) (a) Fid the terms of this fiite sequece. Use the statistical plottig feature of a graphig utility to costruct a bar graph that represets the sequece. (b) What does the patter i the bar graph i part (a) say about the future of the federal debt?

11 _9.qxd // : AM Page Chapter 9 Sequeces, Series, ad Probability. Reveue The reveues (i millios of dollars) for Amazo.com for the years 99 through are show i the figure. The reveues ca be approximated by the model Reveue (i millios of dollars) where is the year, with correspodig to 99. Use this model to approximate the total reveue from 99 through. Compare this sum with the result of addig the reveues show i the figure. (Source: Amazo.com) Sythesis True or False? I Exercises ad, determie whether the statemet is true or false. Justify your aswer.. i i i i. Fiboacci Sequece I Exercises ad, use the Fiboacci sequece. (See Example.). Write the first terms of the Fiboacci sequece a ad the first terms of the sequece give by. Usig the defiitio for b i Exercise, show that ca be defied recursively by Arithmetic Mea I Exercises 7, use the followig defiitio of the arithmetic mea x of a set of measuremets x, x, x,..., x. x x i a b a a, b b.. 7. Fid the arithmetic mea of the six checkig accout balaces $7., $7.9, $., $., $., ad $9.. Use the statistical capabilities of a graphig utility to verify your result. a a.9 9..,, 7,..., 7 9 Year ( 99) j j j j b. Fid the arithmetic mea of the followig prices per gallo for regular uleaded gasolie at five gasolie statios i a city: $.99, $.99, $.99, $.99, ad $.999. Use the statistical capabilities of a graphig utility to verify your result. 9. Proof Prove that. Proof Prove that I Exercises, fid the first five terms of the sequece.. a x!.. a x!. Skills Review I Exercises, determie whether the fuctio has a iverse fuctio. If it does, fid its iverse fuctio.. f x x. gx x 7. hx x. f x x I Exercises 9, fid (a) A B, (b) B A, (c) AB, ad (d) BA I Exercises, fid the determiat of the matrix... A A. A A A. A A A 9, , 7 B 7,, x i x. x i x B B B 7 xi a x a x! x i.