LESSON #66 - SEQUENCES COMMON CORE ALGEBRA II

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1 LESSON #66 - SEQUENCES COMMON CORE ALGEBRA II I Commo Core Algebra I, you studied sequeces, which are ordered lists of umbers. Sequeces are extremely importat i mathematics, both theoretical ad applied. A sequece is formally defied as a fuctio that has as its domai the set the set of positive itegers, i.e., 2, 3,...,. Exercise #: A sequece is defied by the equatio a 2 (a) Fid the first three terms of this sequece, deoted by a(), a (2), ad a(3).. (b) Which term has a value of 53? (c) Explai why there will ot be a term that has a value of 70. Exercise #2: A sequece is defied explicitly as a 3 5. (a) What is the value of the 5 th term i this sequece? (b) What is the value of. (c) With explicit sequece formulas, whe you are lookig for a specific term i the sequece, what do you eed to do? Recall that sequeces ca also be described by usig recursive defiitios. Whe a sequece is defied recursively, terms are foud by operatios o previous terms. Exercise #3: A sequece is defied by the recursive formula: f f 5 with (a) Geerate the first five terms of this sequece. Label each term with proper fuctio otatio. (b) Determie the value of f 2.. Hit thik about how may times you have added 5 to. COMMON CORE ALGEBRA II, UNIT #5 SEQUENCES AND SERIES

2 2 Exercise #4: A sequece is defied recursively as a 2; a 3a. (a) What is the value of the secod term i the sequece? (b) What is the value of the fourth term i the sequece? (c) Whe you are lookig for a specific term i a sequece defied recursively, what must you fid first? Exercise #5: For the recursively defied sequece 2 t t 2 ad t 2, what is the value of t 4? Exercise #6: Oe of the most well-kow sequeces is the Fiboacci, which is defied recursively usig two previous terms. Its defiitio is give below. 2 ad f f f f f ad 2 Geerate values for f 3, f 4, f 5, ad f 6 (i other words, the ext four terms of this sequece). Exercise #7: Which of the followig would represet the graph of the sequece a 2? Explai your choice. y y () (2) (3) (4) y y Explaatio:

3 3 Exercise #8: Match each of the explicit ad recursive formulas with its sequece of umbers. ) Explicit Formula Recursive Formula Sequece f( ) 6( 3) A. f () 6, f ( ) f ( ) 6 5) 6, 9, 2, 5, 2) a 3 3 3) f ( ) a 6, a a, B. 2 6) 6, 3, 2 4 C. f () 6, f ( ) 3 f ( ) 7) 6, 2, 8, 24 4) a 6 2 a 6, a a 3 8) 6, -8, 54, -324 D. ), 3), 2), 4),

4 4 LESSON #66 - SEQUENCES COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Give each of the followig sequeces defied by formulas, determie ad label the first four terms. A variety of differet otatios is used below for practice purposes. (a) f 7 2 (b) a 2 5 (c) t 2 3 (d) t 2. Sequeces below are defied recursively. Determie ad label the ext three terms of the sequece. (a) f 4 ad f f 8 (b) a a a ad 24 2 (c) b b 2 with 5 2 b (d) f 2f ad f 4 4. A recursive sequece is defied by a 2 a a with a 0 ad a2. Which of the followig represets the value of a 5? () 8 (3) 3 (2) 7 (4) 4 5. Which of the followig formulas would represet the sequece 0, 20, 40, 80, 60, a (3) a () 0 (2) 52 a 0 2 (4) a 2 0

5 5-5.

6 6 APPLICATIONS 6. A tilig patter is created from a sigle square ad the expaded as show. If the umber of squares i each patter defies a sequece, the determie the umber of squares i the seveth patter. Explai how you arrived at your choice. Ca you write a recursive defiitio for the patter? REASONING 7. Cosider a sequece defied similarly to the Fiboacci, but with a slight twist: 2 with f f f f f 2 ad 2 5 Geerate terms f 3, f 4, f 5, f 6, f 7, ad f 8. The, determie the value of 25 f.

7 7 LESSON #67 - ARITHMETIC AND GEOMETRIC SEQUENCES COMMON CORE ALGEBRA II I Commo Core Algebra I, you studied two particular sequeces kow as arithmetic (based o costat additio to get the ext term) ad geometric (based o costat multiplyig to get the ext term). I this lesso, we will review the basics of these two sequeces. ARITHMETIC SEQUENCE RECURSIVE DEFINITION Give, the or give where d is called the commo differece ad ca be positive or egative. ARITHMETIC SEQUENCE EXPLICIT FORMULA or where d is called the commo differece ad ca be positive or egative. Exercise #: Cosider f ( ) f ( ) 3 with f() 5. th (a) Determie the value of f (2), f (3), ad f (4). (b) Write a explicit formula for the term of a arithmetic sequece, f( ), based o the first term, f (), d ad. (c) Usig your aswer to (b), fid f(), f(2), f(3), ad f(4) to make sure you foud the correct formula. Exercise #2: Give that a 6 ad a4 8 are members of a arithmetic sequece, (a) Fid the value of d, a 2, ad a 3. (b) Write a recursive defiitio for the the sequece, a. th term of (c) Write a explicit formula for the term of a arithmetic sequece, a. th (c) Determie the value of a. Should you use 20 your aswer to (b) or (c) to do so? Explai.

8 8 Geometric sequeces are defied very similarly to arithmetic sequeces, but with a multiplicative costat istead of a additive oe. GEOMETRIC SEQUENCE RECURSIVE DEFINITION Give or give, the where r is called the commo ratio ad ca be positive or egative ad is ofte fractioal. GEOMETRIC SEQUENCE EXPLICIT FORMULA where r is called the commo ratio ad ca be positive or egative ad is ofte fractioal. or Exercise #3: Cosider a 2 ad a a 3. (a) Geerate the value of. (b) Write a explicit formula for the term of the sequece,, based o the first term,, r, ad. (c) Usig your aswer to (b), fid to make sure you foud the correct formula. Exercise #4: Give that f () 6 ad f (4) 48 are members of a geometric sequece, (a) Fid the value of r, f (2) ad f (3). th (b) Write a recursive defiitio for the term of the sequece, f( ). th (c) Write a explicit formula for the term of a arithmetic sequece, f( ). a. (c) Determie the value of f (6). Should you use your aswer to (b) or (c) to do so? Explai.

9 9 Exercise #5: Complete the chart with the missig iformatio. Note: It is easier to fid the terms first ad work from there. Terms.,3,5,7,9... Arithmetic or Geometric? Explicit Formula Recursive Formula 2. 2,4,8,6,32, f ( ) f ( ) 3 with f() 2 4. (0) a

10 0 FLUENCY LESSON #67 - ARITHMETIC AND GEOMETRIC SEQUENCES COMMON CORE ALGEBRA II HOMEWORK Use the give iformatio to fill i the other three rows i the table. Hit: If the terms are ot give, fid the first few terms before completig the rest of the row. Terms 5. 0,4,8,22... Arithmetic or Geometric? Explicit Formula Recursive Formula 6. 30,5,7.5,3.75, f () = 2 f ( -) with f() = f () = 5+ ( -) 2 a = a with a = a = 3(-4) - 7. Cosider f () = f ( -) -0 with f() = 24. (a) Determie the value of f (2), f (3), ad f (4). (b) Write a explicit formula for the sequece, f( ). th term of the (c) Usig your aswer to (b), fid f(), f(2), f(3), ad f(4) to make sure you foud the correct formula.

11 8. Cosider a = 4 æ ö 3 è ç 3ø -. (a) Determie the value of a,a 2,ad a 3. (b) Write a recursive formula for the sequece. th term of the (c) Usig your aswer to (b), fid a 2 ad a 3 to make sure you foud the correct formula. APPLICATIONS 9. The populatio of Jamesburg for the years , respectively, was reported as follows: 250, ,937 25, ,822 How ca this sequece be recursively modeled? ) 2) 3) 4)

12 2 LESSON #68 SEQUENCE WORD PROBLEMS COMMON CORE ALGEBRA II Exercise #: A store maager plas to offer discouts o some sweaters each week i a attempt to sell them before the witer seaso eds accordig to this sequece: $48, $36, $27, $20.25,... Write a explicit formula ad a recursive formula for this sequece. Exercise #2: After oe customer buys 4 ew tires, a garage recyclig bi has 20 tires i it. After aother customer buys four ew tires, the bi has 24 tires i it. Write a explicit formula to represet the umber of tires i the bi after customers have bought four tires. How may tires would be i the bi after 9 customers buy four ew tires? Exercise #3: A patter exists i the sum of the iterior agles of polygos. The sum of the iterior agles of a triagle is 80, of a quadrilateral is 360, ad of a petago is 540. (a) Which choice is a recursive formula for this patter? () f () = 80; f () = 80 f ( -) (3) f () = 80; f () = f ( -) (2) f () = 80; f () = f ( +) +80 (4) f () = 80; f () = f ( -) +80 (b) What is the sum of the iterior agles of a oago (9 sides)? Note: You ca see that real world situatios ca be defied with recursive ad explicit formulas. I certai situatios it makes sese to iclude 0 i the domai of the fuctio whe there is a startig value at t=0. Exercise #4: A certai culture of yeast icreases by 50% every hour. There are 9 grams of yeast i a culture dish whe the experimet begis. Write a explicit formula ad recursive formula for the growth of the yeast h hours after the experimet begis. Exercise #5: You have a cafeteria card worth $50. After you buy luch o Moday its value is $ After you buy luch o Tuesday, its value is $ Assumig the patter cotiues, write a fuctio rule to represet the amout of moey left o the card. What is the value of the card after you buy 2 luches?

13 3 Exercise #6: Laie has decided to add stregth traiig to her exercise program. Her traier suggests that she add weight liftig for 5 miutes durig her routie the first week. Each week thereafter, she is to icrease the weight liftig time by 2 miutes. Which formula represets this sequetial icrease i weight liftig time? () f () = (3) f () = (2) f () = (4) f () = Exercise #7: The graph to the right models a sequece. a. Does this fuctio show a liear or expoetial relatioship? b. Based o your aswer to the previous questio, fid the commo differece or commo ratio. c. Write a explicit formula for the sequece. d. Write a recursive formula for the sequece. Exercise #8: The graph to the right shows the umber of teams left i the Wome s Basketball touramet at the begiig of each roud. a. Does this fuctio show a liear or expoetial relatioship? b. Based o your aswer to the previous questio, fid the commo differece or commo ratio. c. Write a explicit formula for the sequece. d. Write a recursive formula for the sequece.

14 4 Exercise #9: The formula below ca be used to model which sceario? ) The first row of a stadium has 3000 seats, ad each row thereafter has 80 more seats tha the row i frot of it. 2) The last row of a stadium has 3000 seats, ad each row before it has 80 fewer seats tha the row behid it. 3) A bak accout starts with a deposit of $3000, ad each year it grows by 80%. 4) The iitial value of a specialty toy is $3000, ad its value each of the followig years is 20% less. Exercise #0: The recursive formula below ca be used to model the amout of moey i Rebecca s bak accout. Explai what iformatio is give by the formula. a 6500 a a (.042) Exercise #: The value of a car i the years , respectively were reported as follows: 30,000 28, (a) Write a recursive formula to model the cost of the car. (b) If the patter cotiues, use the recursive formula to fid the value of the car i 209 to the earest dollar.

15 5 LESSON #68 - SEQUENCE WORD PROBLEMS COMMON CORE ALGEBRA II HOMEWORK. Suppose you are rehearsig for a cocert. You pla to rehearse the piece you will perform four times the first day ad the to double the umber of times you rehearse the piece each day util the cocert. a. Write a sequece of umbers to represet this situatio. b. Is the sequece arithmetic, geometric, or either? c. Write a explicit formula ad a recursive formula for this sequece. d. If the cocert was i days, how may times would you rehearse the piece o the 0 th day? Is this reasoable? 2. Write a recursive formula ad a explicit formula for the sequece modeled i the graph to the right. (Assume the first poit o the graph is (,-3)). 3. A Greek theater has 30 seats i the first row of the ceter sectio. Each row behid the first row gais two additioal seats. How may seats are i the 25 th row? Write a explicit formula for this situatio to fid your aswer. 4. A research lab is to begi experimetatio with bacteria that grows by 20% each hour. The lab has 200 bacteria to begi the experimet. a. Write a recursive formula ad a explicit formula to model the umber of bacteria after hours. b. How may whole bacteria will be preset at the ed of the 2 th hour?

16 6 5. The summer Olymics occur every four years. a) Which formula represets the years of the summer Olympics, startig with 206? ) f()=206+4(+) 2) f()= ) f()=206+4(-) 4) f()=206+(+4) 6. Mr. Carlso suffers from allergies. Whe allergy seaso arrives, his doctor recommeds that he take 300 mg of his medicatio the first day ad decrease the dosage by oe half each day for oe week. a. Which rule represets his medicatio doses for the week? æ () f () = 300 ö è ç 2ø - (3) f () = æ (2) f () = ö è ç 2ø - (4) f () = 2 b. To the earest milligram, what is the amout of medicatio Mr. Carlso will take o the 7 th day? 7. Your father wats you to help him build a shed i the backyard. He says he will pay you $50 for the first week ad 5% more each week after that. a. Write a explicit formula ad a recursive defiitio to model this situatio. b. How much will your father pay you i the 5 th week to the earest cet? 8. Your gradmother gives you $000 to start a college book fud. She tells you she will add $200 to the fud each moth after that, if you will add $5 each moth. a. Which "rule" geerates a sequece of the mothly amouts i your college book fud? ) f()= (-) 2) f()= ) f () = 000; f () = f ( - ) ) All three formulas geerate this sequece. b. After how may moths will the college book fud have $575?

17 7 9. The formula below ca be used to model which sceario? a 6700 a.675 ) A perso had $6700 i the first moth of the year ad made $67.50 each moth after that. 2) A perso had $6700 i the first moth of the year, ad that amout icreased 67.5% each moth after that. 3) A perso had $6700 i the first moth of the year, ad that amout decreased 32.5% each moth after that. 4) A perso had $6700 i the first moth of the year, ad that amout decreased 67.5% each moth after that. a 0. The umber of bacteria i a sample for four cosecutive hours are listed below (a) Write a recursive formula to model the umber of bacteria i the sample. (b) If the patter cotiues, use the recursive formula to fid the umber of bacteria the ext two hours.

18 8 LESSON #69 - SUMMATION NOTATION COMMON CORE ALGEBRA II Much of our work i this uit will cocer addig the terms of a sequece. I order to specify this additio or summarize it, we itroduce a ew otatio, kow as summatio or sigma otatio that will represet these sums. This otatio will also be used later i the course whe we wat to write formulas used i statistics. SUMMATION (SIGMA) NOTATION where i is called the idex variable, which starts at a value of a, eds at a value of, ad moves by uit icremets (icrease by each time). Exercise #: Evaluate each of the followig sums. (a) 5 2i (b) i k (c) 32 j k j 5 (d) i (e) 2( k ) (f) ii i 4 k 3 i Exercise #2: Which of represets the value of () 0 (3) i? i (2) 9 4 (4) 3 24

19 9 Exercise #3: Cosider the sequece defied recursively by a a 2 a2 ad a 0 ad a2. Fid the value of 7 i4 a i Exercise #4: It is also good to be able to place sums ito sigma otatio. The values that are beig summed i the ext problems form either a arithmetic or geometric sequece. Look back at Exercise # o the previous page. Which problem represeted the sum of the terms i a arithmetic sequece? A geometric sequece? Exercise #5: Express each sum usig sigma otatio. Use as your idex variable. First determie if the sequece is arithmetic or geometric. Secod, determie, the umber of terms. The use the appropriate explicit formula to write the sum. (a) (b) (c) (d) Exercise #6: Some sums are more iterestig tha others. Determie the value of reasoig. This is kow as a telescopig series (or sum). 99. Show your i i i

20 20 LESSON #69 - SUMMATION NOTATION COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Evaluate each of the followig. Place ay o-iteger aswer i simplest ratioal form. (a) i (b) k i2 k 0 (c) 4 (d) 3 k k (e) 2 2 k 3 (f) log 0 i k 0 i 2. Which of the followig is the value of 4k () 53 (3) 37 (2) 45 (4) 80 4? k 0 3. The sum () i is equal to i4 (3) 3 4 (2) 3 2 (4) Which of the followig represets the sum ? (3) j j j () 4j 3 03 (4) 4 j (2) j 2 j3 j0

21 2 5. Express each sum usig sigma otatio. Use as your idex variable. (a) (b) (c) for 20 terms (d) A sequece is defied recursively by the formula b 4b 2 b 2 with b ad b2 3. What is the value of 5 bi? Show the work that leads to your aswer. i3 REASONING 7. A curious patter occurs whe we look at the behavior of the sum 2k (a) Fid the value of this sum for a variety of values of below: 2 k 4 : 2k 2 : 2 k 4 k. k 3 k 5: 2k 3: 2 k 5 k (b) What types of umbers are you summig? What types of umbers are the sums? (c) Fid the value of such that.

22 22 LESSON #70 - GEOMETRIC SERIES COMMON CORE ALGEBRA II A series is simply the sum of the terms of a sequece. The fudametal defiitio/otio of a series is below. THE DEFINITION OF A SERIES If the set represet the elemets of a sequece the the series,, is defied by: I truth, you have already worked extesively with series i previous lessos almost aytime you evaluated a summatio problem. Exercise #: Give a geometric series defied by the recursive formula a 3 ad a a 2, which of the followig is the value of S 5 5 a? i () 06 (3) 93 (2) 75 (4) 35 i Exercise #3: Which of the followig represets the sum of a geometric series with 8 terms whose first term is 3 ad whose commo ratio is 4? () 32,756 (3) 42,560 (2) 28,765 (4) 65,535 SUM OF A FINITE GEOMETRIC SERIES For a geometric series defied by its first term,, ad its commo ratio, r, the sum of terms is give by: or Exercise #4: Fid the value of the geometric series show below. Show the calculatios that lead to your fial aswer

23 23 Exercise #5: Maria saved $500 the first year after college ad icreased the amout she saved 5% each year after that. (a) Write a geometric series formula, S, for Maria s total savigs over years. (b) Use this formula to fid Maria s total savigs for the first years after college. Exercise #6: Justi wats to save moey after fiishig college. He also starts by savig $500 the first year after college, but he saves 5% less each year after that. (a) Write a geometric series formula, S, for Justi s total savigs over years. (b) Use this formula to fid Justi s total savigs for the first years after college. (c) How much more moey has Maria (Exercise #6) saved tha Justi i the first years after college. Exercise #7: A perso places pey i a piggy bak o the first day of the moth, 2 peies o the secod day, 4 peies o the third, ad so o. Will this perso be a millioaire at the ed of a 3 day moth? Show the calculatios that lead to your aswer.

24 24 LESSON #70 - GEOMETRIC SERIES COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Fid the sums of geometric series with the followig properties: (a) a 6, r 3 ad 8 (b) a 20, r, ad 6 (c) a 2 5, r 2, ad 0 2. If the geometric series () (2) has seve terms i its sum the the value of the sum is 27 (3) (4) A geometric series has a first term of 32 ad a fial term of this series is () 9.75 (3) 22.5 (2) 6.25 (4) Which of the followig represets the value of 8 i0 has i it. () 9,7 (3) 22,34 (2) 2,60 (4) 8,956 ad a commo ratio of 4. The value of 2 i 3 256? Thik carefully about how may terms this series 2 5. A geometric series whose first term is 3 ad whose commo ratio is 4 sums to The umber of terms i this sum is () 8 (3) 6 (2) 5 (4) 4

25 25 6. Fid the sum of the geometric series show below. Show the work that leads to your aswer APPLICATIONS 7. I the picture show at the right, the outer most square has a area of 6 square iches. All other squares are costructed by coectig the midpoits of the sides of the square it is iscribed withi. Fid the sum of the areas of all of the squares show. First, cosider the how the area of each square relates to the larger square that surrouds (circumscribes) it. 8. Maraly made $42,000 the first year at her job ad she received a 2.5% raise each year. (a) Write a geometric series formula, S, for the total amout Maraly has made over years. (b) Use this formula to fid total earigs for the first 20 years at the job. 9. Jeifer fids a way to do her math homework more efficietly. The first ight it takes her 32 miutes. Each ight after that, it takes her 3 4 of the time it did the previous ight. Was the total amout of time, to the earest miute, she took doig her homework i the first seve ights? Write out the first five terms of this sum to help visualize.

26 26 LESSON #7 - MORTGAGE PAYMENTS COMMON CORE ALGEBRA II Mothly mortgage paymets ca be foud usig the formula below. This formula comes from a geometric series, but we will just be learig how to work with the formula ad solve for the differet variables. M r r P 2 2 r 2 M = mothly paymet P = amout borrowed r = aual iterest rate = umber of mothly paymets The most basic way to use the formula is to calculate mothly paymets. Exercise #: You took out a 30-year mortgage for $220,000 to buy a house. The iterest rate o the mortgage is 5.2%. a. What are your mothly paymets to the earest dollar? b. With this mothly paymet, what is the total cost to pay off the loa? Whe you are takig out a mortgage, you ofte kow how much you ca afford each moth, ad you wat to determie what size mortgage you ca afford. Exercise #2: Based o your curret icome, you ca afford mortgage paymets of $900 a moth. You also wat to take out a 5 year mortgage to pay off the loa sooer. If the average iterest rate at this time is 3.375%, what size mortgage ca you afford to the earest cet?

27 27 M r r P 2 2 r 2 M = mothly paymet P = amout borrowed r = aual iterest rate = umber of mothly paymets The last way we will lear to use the mortgage paymet formula ivolves determiig the legth of the loa that you ca afford give the cost of a house, the amout you ca sped per moth, ad the iterest rate. Exercise #3: Imagie you have foud the house of your dreams for $325,000. You kow you ca afford mothly mortgage paymets of $500. You qualified for a mortgage with a iterest rate of 4.75%. Algebraically determie the umber of paymets you would eed to make to pay off the loa at this rate to the earest whole umber. How may years would it take you to the earest teth of a year? Exercise #4: You are iterested i purchasig a codo that costs $59,000. You kow you ca afford mothly mortgage paymets of $900. You qualified for a mortgage with a iterest rate of 3.825%. Algebraically determie the umber of paymets you would eed to make to pay off the loa at this rate to the earest whole umber. How may years would it take you to the earest teth of a year?

28 28 Extra Practice Exercise #5: You took out a 20-year mortgage for $80,000 to buy a house. The iterest rate o the mortgage is 4.8%. a. What are your mothly paymets to the earest dollar? b. With this mothly paymet, what is the total cost to pay off the loa? Exercise #6: Based o your curret icome, you ca afford mortgage paymets of $00 a moth. You also wat to take out a 30-year mortgage to spread the paymets out over time. If the average iterest rate at this time is 5.25%, what size mortgage ca you afford to the earest dollar? Exercise #7: You have chose a starter home for $20,000. You kow you ca afford mothly mortgage paymets of $700. You qualified for a mortgage with a iterest rate of 3.25%. Algebraically determie the umber of paymets you would eed to make to pay off the loa at this rate to the earest whole umber. How may years would it take you to the earest teth of a year?

29 29 LESSON #7 - MORTGAGE PAYMENTS COMMON CORE ALGEBRA II HOMEWORK. You took out a 5-year mortgage for $60,000 to buy a house. The iterest rate o the mortgage is 5.2%. What are your mothly paymets to the earest cet? 2. Based o your curret icome, you ca afford mortgage paymets of $250 a moth. You also wat to take out a 25-year mortgage to spread the paymets out over time. If the average iterest rate at this time is 3.375%, what size mortgage ca you afford to the earest dollar? 3. You have chose a home i the perfect locatio for $250,000. You kow you ca afford mothly mortgage paymets of $,400. You qualified for a mortgage with a iterest rate of 4.75%. Algebraically determie the umber of paymets, to the earest whole paymet, you would eed to make to pay off the loa at this rate. How may years would it take you to the earest teth of a year?

30 30 4. You took out a 30-year mortgage for $350,000 to buy a house. The iterest rate o the mortgage is 4.5%. a. What are your mothly paymets to the earest dollar? b. With this mothly paymet, what is the total cost to pay off the loa? 5. You foud a small home for $30,000. You kow you ca afford mothly mortgage paymets of $990. You qualified for a mortgage with a iterest rate of 4.0%. Algebraically determie the umber of paymets, to the earest whole paymet, you would eed to make to pay off the loa at this rate. How may years would it take you to the earest teth of a year?