Sequences and Series. Growth of $5,000 investment per year. Amount (in thousands of dollars) 150. Time (years)

Transcription

1 C H A P T E R Sequeces ad Series Amout (i thousads of dollars) Growth of $5,000 ivestmet per year Time (years) veryoe realizes the importace of ivestig for the future. Some people go to great pais to study the markets ad to make E wise ivestmet decisios. Some stay away from ivestig because they do ot wat to take chaces. However, the most importat factor i ivestig is makig regular ivestmets (Moey, Accordig to Moey, if you had ivested $5,000 i the stock market every year at the market high for that year (the worst time to ivest) for the last 40 years, your ivestmet would be worth $.8 millio today. A sequece of periodic ivestmets earig a fixed rate of iterest ca be thought of as a geometric sequece. I this chapter you will lear how to fid the sum of a geometric sequece ad to calculate the future value of a sequece of periodic ivestmets. I Exercise 58 of Sectio.4 you will see how Moey magazie calculated the value of $5,000 ivested each year for 0 years i Fidelity s Magella fud.

2 68 (-) Chapter Sequeces ad Series. SEQUENCES I this sectio The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is out of sequece. I this sectio we give the mathematical defiitio of a sequece. Defiitio Fidig a Formula for the th Term Defiitio I mathematics we thik of a sequece as a list of umbers. Each umber i the sequece is called a term of the sequece. There is a first term, a secod term, a third term, ad so o. For example, the daily high temperature readigs i Miot, North Dakota, for the first 0 days i Jauary ca be thought of as a fiite sequece with 0 terms: 9,, 8,,0,6,4,,5, The set of all positive eve itegers,, 4, 6, 8, 0,, 4,..., ca be thought of as a ifiite sequece. To give a precise defiitio of sequece, we use the termiology of fuctios. The list of umbers is the rage of the fuctio. Sequece A fiite sequece is a fuctio whose domai is the set of positive itegers less tha or equal to some fixed positive iteger. A ifiite sequece is a fuctio whose domai is the set of all positive itegers. Whe the domai is apparet, we will refer to either a fiite sequece or a ifiite sequece simply as a sequece. For the idepedet variable of the fuctio we will usually use (for atural umber) rather tha x. For the depedet variable we write a (read a sub ) rather tha y. We call a the th term, or the geeral term of the sequece. Rather tha use the f (x) otatio for fuctios, we will defie sequeces with formulas. Whe is used as a variable, we will assume it represets atural umbers oly. E X A M P L E Listig terms of a fiite sequece List all of the terms of each fiite sequece. a) a for 5 b) a for 4 Solutio a) Usig the atural umbers from through 5 i a, we get a, a 4, a 3 3 9, a 4 4 6,

3 . Sequeces (-3) 69 calculator close-up Because a sequece is a fuctio whose domai is the set of positive itegers, we ca defie the sequece with the Y key ad make a list of the terms. ad a The five terms of this sequece are, 4, 9, 6, ad 5. We ofte refer to the listig of the terms of the sequece as the sequece. b) Usig the atural umbers from through 4 i a, we get the terms a, 3 a, 4 a 3, 5 ad a 4. 6 The four terms of the sequece are 3, 4, 5, ad 6. E X A M P L E calculator Listig terms of a ifiite sequece List the first three terms of the ifiite sequece whose th term is () a. close-up Some calculators have a sequece feature that allows you to specify the formula ad which terms to evaluate. We ca eve get the terms as fractios. Solutio Usig the atural umbers,, ad 3 i the formula for the th term yields () () () 3 a, a, ad a We write the sequece as follows: 4, 8,,... 6 Fidig a Formula for the th Term We ofte kow the terms of a sequece ad wat to write a formula that will produce those terms. To write a formula for the th term of a sequece, examie the terms ad look for a patter. Each term is a fuctio of the term umber. The first term correspods to, the secod term correspods to, ad so o. E X A M P L E 3 A familiar sequece Write the geeral term for the ifiite sequece 3, 5, 7, 9,,.... Solutio The eve umbers are all multiples of ad ca be represeted as. Because each odd umber is more tha a eve umber, a formula for the th term might be a.

4 630 (-4) Chapter Sequeces ad Series helpful hit Fidig a formula for a sequece could be extremely difficult. For example, there is o kow formula that will produce the sequece of prime umbers:, 3, 5, 7,, 3, 7, 9,... To be sure, we write out a few terms usig the formula: a () 3 a () 5 a 3 (3) 7 So the geeral term is a. CAUTION There ca be more tha oe formula that produces the give terms of a sequece. For example, the sequece,, 4,... could have th term a or a. The first three terms for both of these sequeces are idetical, but their fourth terms are differet. E X A M P L E 4 A sequece with alteratig sigs Write the geeral term for the ifiite sequece, 4,,, Solutio To obtai the alteratig sigs, we use powers of. Because ay eve power of is positive ad ay odd power of is egative, we use (). The deomiators are the squares of the positive itegers. So the th term of this ifiite sequece is give by the formula () a. Check this sequece by usig this formula to fid the first four terms. I the ext example we use a sequece to model a physical situatio. E X A M P L E 5 6 ft 4 ft 8 ft 3 The boucig ball Suppose a ball always rebouds 3 of the height from which it falls ad the ball is dropped from a height of 6 feet. Write a sequece whose terms are the heights from which the ball falls. What is a formula for the th term of this sequece? Solutio O the first fall the ball travels 6 feet (ft), as show i Fig... O the secod fall it travels 8 of 6, or 4 ft. O the third fall it travels of 4, or ft, ad so o. We write the sequece as follows: 8 6 6, 4,,, 3, FIGURE. The th term ca be writte by usig powers of 3 : 3 a 6

5 . Sequeces (-5) 63 M A T H A T W O R K Most of us fid a upholstered chair, sik ito it, ad remark o the comfort. Before desig cosultat Audrey Jorda sits dow, she ofte looks at the fabric to observe the color ad texture ad especially to see whether the fabric is oe of her origial desigs. Fabric desig is more tha just a idea that is prited o a piece of cloth. Cosideratio must be give to the ed product, which could be aythig from a hadbag to a large sofa. Colors ad themes must be chose with both curret treds ad styles i mid. Sometimes a desig will be a overall or odirectioal patter, such as polka dots, which ca be cut radomly. More ofte, it will have a specific theme, such as fruit, which ca be cut ad sew i oly oe directio. For all products oe of the mai cosideratios is the vertical repeat. A good portio of textile machiery is stadardized for vertical repeats of 7 iches or fractios thereof. For example, the vertical repeat could be every 3 iches or every 9 iches. Eve though the horizotal repeat ca vary, Ms. Jorda must cosider both the horizotal ad vertical repeats for a particular ed product. I Exercise 45 of this sectio you will fid the stadard vertical repeats for a textile machie. FABRIC DESIGNER WARM-UPS True or false? Explai your aswer.. The th term of the sequece, 4, 6, 8, 0,... is a. True. The th term of the sequece, 3, 5, 7, 9,... is a. True 3. A sequece is a fuctio. True 4. The domai of a fiite sequece is the set of positive itegers. False 5. The th term of, 4, 9, 6, 5,... is a (). False 6. For the ifiite sequece b, the idepedet variable is. False 7. For the sequece c 3, the depedet variable is c. True 8. The sixth term of the sequece a () is 64. False 9. The symbol a is used for the depedet variable of a sequece. True 0. The teth term of the sequece, 4, 8, 6, 3, 64, 8,... is 04. True. EXERCISES Readig ad Writig After readig this sectio, write out the aswers to these questios. Use complete seteces.. What is a sequece? A sequece is a list of umbers.. What is a term of a sequece? Each umber i the sequece is called a term of the sequece. 3. What is a fiite sequece? A fiite sequece is a fuctio whose domai is the set of positive itegers less tha or equal to some fixed positive iteger. 4. What is a ifiite sequece? A ifiite sequece is a fuctio whose domai is the set of all positive itegers.

6 63 (-6) Chapter Sequeces ad Series List all terms of each fiite sequece. See Example. 5. a for 8, 4, 9, 6, 5, 36, 49, a for 4, 4, 9, 6 () 7. b for 0,, 3, 4, 5, 6, 7, 8, 9, 0 () 8. b for 6,, 3, 4, 5, 6 9. c () for 5,, 4, 8, 6 0. c (3) for 5,, 3, 9, 7 3. a for 6, 4, 8, 6,, a for 5,,, 4, 8 3. b 3 for 7,, 3, 5, 7, 9, 4. b 6 for 7 8, 0,, 4, 6, 8, 0 5. c for 5,, 3, 3, c for 4,, 3, Write the first four terms of the ifiite sequece whose th term is give. See Example. 7. a 8. b ( )( ), 6,, 0 6,,, b 0. a 5 5 3,,, 3 7 4, 4 9, 4 4, 3. c () ( ). c () ( ), 0,, 4, 9, 5, 49 () 3. a 4. a (), 4, 9, 6,, 4, 8 Write a formula for the geeral term of each ifiite sequece. See Examples 3 ad 4. 5., 3, 5, 7, 9,... a 6. 5, 7, 9,, 3,... a 3 7.,,,,... a () 8.,,,,... a () 9. 0,, 4, 6, 8,... a 30. 4, 6, 8, 0,,... a 3. 3, 6, 9,,... a , 8,, 6,... a , 7, 0, 3,... a , 7,, 5,... a 4 35.,, 4, 8, 6,... a () 36., 3, 9, 7,... a (3) 37. 0,, 4, 9, 6,... a ( ) 38. 0,, 8, 7, 64,... a ( ) 3 Solve each problem. See Example Football pealties. A football is o the 8-yard lie, ad five pealties i a row are give that move the ball half the distace to the (closest) goal. Write a sequece of five terms that specify the locatio of the ball after each pealty. 4,,,, yard lie Ifestatio. Leoa plated 9 acres of soybeas, but by the ed of each week, isects had destroyed oe-third of the acreage that was healthy at the begiig of the week. How may acres does she have left after 6 weeks? 64 acre 8 4. Costat rate of icrease. The MSRP for the 999 Ford F-50 Lariat 4WD Super Duty Super Cab was $3,535 MSRP (i thousads of dollars) Ford F-50 4WD Super Cab Model year FIGURE FOR EXERCISE 4

7 . Sequeces (-7) 633 (Edmud s New Car Prices, Suppose the price of this truck icreases by 5% each year. Fid the prices to the earest dollar for the 000 through 005 models. $34,6, $35,870, $37,663, $39,546, $4,54, $43, Costat icrease. The MSRP for a ew 999 Mercury Cougar was $,455 (Edmud s New Car Prices, www. edmuds.com). Suppose the price of this car icreases by $,000 each year. Fid the prices of the 000 through 005 models. $,455, $3,455, $4,455, $5,455, $6,455, $7, Ecoomic impact. To assess the ecoomic impact of a factory o a commuity, ecoomists cosider the aual amout the factory speds i the commuity, the the portio of the moey that is respet i the commuity, the the portio of the respet moey that is respet i the commuity, ad so o. Suppose a garmet maufacturer speds $ millio aually i its commuity ad 80% of all moey received i the commuity is respet i the commuity. Fid the first four terms of the ecoomic impact sequece. $,000,000, $800,000, $640,000, $5,000 Amout (i millios of dollars) % rate Number of respedigs FIGURE FOR EXERCISE Less impact. The rate at which moey is respet i a commuity varies from commuity to commuity. Fid the first four terms of the ecoomic impact sequece for the maufacturer i Exercise 43, assumig oly 50% of moey received i the commuity is respet i the commuity. $,000,000, $500,000, $50,000, $5, Fabric desig. A fabric desiger must take ito accout the capability of textile machies to produce material with vertical repeats. A textile machie ca be set up for a vertical repeat every 7 iches (i.), where is a atural umber. Write the first five terms of the sequece a 7, which gives the possible vertical repeats for a textile machie. 7 i., 3.5 i., 9 i., 6.75 i., 5.4 i. 46. Musical toes. The ote middle C o a piao is tued so that the strig vibrates at 6 cycles per secod, or 6 Hertz (Hz). The C ote oe octave higher is tued to 54 Hz. The tuig for the otes i betwee usig the method called equal temperamet is determied by the sequece a 6. Fid the tuig for the otes i betwee. 78, 94, 3, 330, 350, 37, 393, 46, 44, 467, 495 Hz. GETTING MORE INVOLVED 47. Discussio. Everyoe has two (biological) parets, four gradparets, eight great-gradparets, 6 great-greatgradparets, ad so o. If we put the word great i frot of the word gradparets 35 times, the how may of this type of relative do you have? Is this more or less tha the preset populatio of the earth? Give reasos for your aswers. 37,438,953,500, larger 48. Discussio. If you deposit cet ito your piggy bak o September ad each day thereafter deposit twice as much as o the previous day, the how much will you be depositig o September 30? The total amout deposited for the moth ca be foud without addig up all 30 deposits. Look at how the amout o deposit is icreasig each day ad see whether you ca fid the total for the moth. Give reasos for your aswers. $5,368,709., $0,737, Cooperative learig. Workig i groups, have someoe i each group make up a formula for a, the th term of a sequece, but do ot show it to the other group members. Write the terms of the sequece o a piece of paper oe at a time. After each term is give, ask whether ayoe kows the ext term. Whe the group ca correctly give the ext term, ask for a formula for the th term. 50. Exploratio. Fid a real-life sequece i which all of the terms are the same. Fid oe i which each term after the first is oe larger tha the previous term. Fid out what the sequece of fies is o your campus for your first, secod, third, ad fourth parkig ticket. 5. Exploratio. Cosider the sequece whose th term is a (0.999). a) Calculate a 00, a 000, ad a 0, , , b) What happes to a as gets larger ad larger? a goes to zero