Accelerated Mathematics III Frameworks Studet Editio Uit Sequeces ad Series d Editio April, 011
Table of Cotets INTRODUCTION:... 3 Reaissace Festival Learig Task... 8 Fasciatig Fractals Learig Task... 17 Divig ito Diversios Learig Task... 8 April, 011 Page of 34
Accelerated Mathematics III Uit Sequeces ad Series Studet Editio INTRODUCTION: I earlier grades, studets leared about arithmetic ad geometric sequeces ad their relatioships to liear ad expoetial fuctios, respectively. This uit builds o studets uderstadigs of those sequeces ad exteds studets kowledge to iclude arithmetic ad geometric series, both fiite ad ifiite. Summatio otatio ad properties of sums are also itroduced. Additioally, studets will examie other types of sequeces ad, if appropriate, proof by iductio. They will use their kowledge of the characteristics of the types of sequeces ad the correspodig fuctios to compare scearios ivolvig differet sequeces. ENDURING UNDERSTANDINGS: All arithmetic ad geometric sequeces ca be expressed recursively ad explicitly. Some other sequeces also ca be expressed i both ways but others caot. Arithmetic sequeces are idetifiable by a commo differece ad ca be modeled by liear fuctios. Ifiite arithmetic series always diverge. Geometric sequeces are idetifiable by a commo ratio ad ca be modeled by expoetial fuctios. Ifiite geometric series diverge if r 1 ad coverge is r 1. The sums of fiite arithmetic ad geometric series ca be computed with easily derivable formulas. Idetifiable sequeces ad series are foud i may aturally occurrig objects. Repeatig decimals ca be expressed as fractios by summig appropriate ifiite geometric series. The priciple of mathematical iductio is a method for provig that a statemet is true for all positive itegers (or all positive itegers greater tha a specified iteger). KEY STANDARDS ADDRESSED: MA3A9. Studets will use sequeces ad series a. Use ad fid recursive ad explicit formulae for the terms of sequeces. b. Recogize ad use simple arithmetic ad geometric sequeces. c. Ivestigate limits of sequeces. d. Use mathematical iductio to fid ad prove formulae for sums of fiite series. April, 011 Page 3 of 34
e. Fid ad apply the sums of fiite ad, where appropriate, ifiite arithmetic ad geometric series. f. Use summatio otatio to explore series. g. Determie geometric series ad their limits. RELATED STANDARDS ADDRESSED: MA3A1. Studets will explore ratioal fuctio. a. Ivestigate ad explore characteristics of ratioal fuctios, icludig domai, rage, zeros, poits of discotiuity, itervals of icrease ad decrease, rates of chage, local ad absolute extrema, symmetry, asymptotes, ad ed behavior. MA3A4. Studets will ivestigate fuctios. a. Compare ad cotrast properties of fuctios withi ad across the followig types: liear, quadratic, polyomial, power, ratioal, expoetial, logarithmic, trigoometric, ad piecewise. b. Ivestigate trasformatios of fuctios. MA3P1. Studets will solve problems (usig appropriate techology). a. Build ew mathematical kowledge through problem solvig. b. Solve problems that arise i mathematics ad i other cotexts. c. Apply ad adapt a variety of appropriate strategies to solve problems. d. Moitor ad reflect o the process of mathematical problem solvig. MA3P. Studets will reaso ad evaluate mathematical argumets. a. Recogize reasoig ad proof as fudametal aspects of mathematics. b. Make ad ivestigate mathematical cojectures. c. Develop ad evaluate mathematical argumets ad proofs. d. Select ad use various types of reasoig ad methods of proof. MA3P3. Studets will commuicate mathematically. a. Orgaize ad cosolidate their mathematical thikig through commuicatio. b. Commuicate their mathematical thikig coheretly ad clearly to peers, teachers, ad others. c. Aalyze ad evaluate the mathematical thikig ad strategies of others. d. Use the laguage of mathematics to express mathematical ideas precisely. MA3P4. Studets will make coectios amog mathematical ideas ad to other disciplies. a. Recogize ad use coectios amog mathematical ideas. b. Uderstad how mathematical ideas itercoect ad build o oe aother to produce a coheret whole. c. Recogize ad apply mathematics i cotexts outside of mathematics. April, 011 Page 4 of 34
MA3P5. Studets will represet mathematics i multiple ways. a. Create ad use represetatios to orgaize, record, ad commuicate mathematical ideas. b. Select, apply, ad traslate amog mathematical represetatios to solve problems. c. Use represetatios to model ad iterpret physical, social, ad mathematical pheomea. UNIT OVERVIEW: The lauchig activity begis by revisitig ideas of arithmetic sequeces studied i eighth ad ith grades. Defiitios, as well as the explicit ad recursive forms of arithmetic sequeces are reviewed. The task the itroduces summatios, icludig otatio ad operatios with summatios, ad summig arithmetic series. The secod set of tasks reviews geometric sequeces ad ivestigates sums, icludig ifiite ad fiite geometric series, i the cotext of explorig fractals. It is assumed that studets have some level of familiarity with geometric sequeces ad the relatioship betwee geometric sequeces ad expoetial fuctios. The third group addresses some commo sequeces ad series, icludig the Fiboacci sequece, sequeces with factorials, ad repeatig decimals. Additioally, mathematical iductio is employed to prove that the explicit forms are valid. The culmiatig task is set i the cotext of applyig for a job at a iterior desig agecy. I each task, studets will eed to determie which type of sequece is called for, justify their choice, ad occasioally prove they are correct. Studets will complete the hadshake problem, the salary/retiremet pla problem, ad some ope-eded desig problems that require the use of various sequeces ad series. VOCABULARY AND FORMULAS Arithmetic sequece: A sequece of terms a1, a, a 3,... with d a a 1. The explicit formula is give by a a1 1 d ad the recursive form is a 1 = value of the first term ad a a d. 1 Arithmetic series: The sum of a set of terms i arithmetic progressio a1 a a3... with d a a. 1 Commo differece: I a arithmetic sequece or series, the differece betwee two cosecutive terms is d, d a a 1. April, 011 Page 5 of 34
Commo ratio: I a geometric sequece or series, the ratio betwee two cosecutive terms is r, a r. a 1 Explicit formula: A formula for a sequece that gives a direct method for determiig the th term of the sequece. It presets the relatioship betwee two quatities, i.e. the term umber ad the value of the term. Factorial: If is a positive iteger, the otatio! (read factorial ) is the product of all positive itegers from dow through 1; that is,! 1... 3 1. Note that 0!, by defiitio, is 1; i.e. 0! 1. Fiite series: A series cosistig of a fiite, or limited, umber of terms. Ifiite series: A series cosistig of a ifiite umber of terms. Geometric sequece: A sequece of terms a1, a, a 3,... with give by 1 1 a a1r ad the recursive form is 1 value of the first term a a April, 011 Page 6 of 34 r. The explicit formula is a ad a a 1 r. Geometric series: The sum of a set of terms i geometric progressio a1 a a3... with a r. a 1 Limit of a sequece: The log-ru value that the terms of a coverget sequece approach. Partial sum: The sum of a fiite umber of terms of a ifiite series. Recursive formula: Formula for determiig the terms of a sequece. I this type of formula, each term is depedet o the term or terms immediately before the term of iterest. The recursive formula must specific at least oe term precedig the geeral term. Sequece: A sequece is a ordered list of umbers. Summatio or sigma otatio: i 1 a, where i is the idex of summatio, is the upper limit of i summatio, ad 1 is the lower limit of summatio. This expressio gives the partial sum, the sum of the first terms of a sequece. More geerally, we ca write value. i k a, where k is the startig Sum of a fiite arithmetic series: The sum, S, of the first terms of a arithmetic sequece is a a give by S, where a 1 = value of the first term ad a = value of the last term i the sequece. 1 i
Sum of a fiite geometric series: The sum, S, of the first terms of a geometric sequece is give by S a1 a a1 1 r 1 r 1 r r, where a 1 is the first term ad r is the commo ratio (r 1). Sum of a ifiite geometric series: The geeral formula for the sum S of a ifiite geometric a1 series a1 a a 3... with commo ratio r where r 1 is S. If a ifiite geometric 1 r series has a sum, i.e. if r 1, the the series is called a coverget geometric series. All other geometric (ad arithmetic) series are diverget. Term of a sequece: Each umber i a sequece is a term of the sequece. The first term is geerally oted as a 1, the secod as a,, the th term is oted as a. a is also referred to as the geeral term of a sequece. April, 011 Page 7 of 34
RENAISSANCE FESTIVAL LEARNING TASK As part of a class project o the Reaissace, your class decided to pla a reaissace festival for the commuity. Specifically, you are a member of differet groups i charge of plaig two of the cotests. You must help pla the archery ad rock throwig cotests. The followig activities will guide you through the plaig process. Group Oe: Archery Cotest 1 Before plaig the archery cotest, your group decided to ivestigate the characteristics of the target. The target beig used has a ceter, or bull s-eye, with a radius of 4 cm, ad ie rigs that are each 4 cm wide. 1. The Target a. Sketch a picture of the ceter ad first 3 rigs of the target. b. Write a sequece that gives the radius of each of the cocetric circles that comprise the etire target. c. Write a recursive formula ad a explicit formula for the terms of this sequece. d. What would be the radius of the target if it had 5 rigs? Show how you completed this problem usig the explicit formula. e. I the past, you have studied both arithmetic ad geometric sequeces. What is the differece betwee these two types of sequeces? Is the sequece i (b) arithmetic, geometric, or either? Explai. Oe versio of the explicit formula uses the first term, the commo differece, ad the umber of terms i the sequece. For example, if we have the arithmetic sequece, 5, 8, 11, 14,, we see that the commo differece is 3. If we wat to kow the value of the 0 th term, or a 0, we could thik of startig with a 1 = ad addig the differece, d = 3 a certai umber of times. How may times would we eed to add the commo differece to get to the 0 th term? Because multiplicatio is repeated additio, istead of addig 3 that umber of times, we could multiply the commo differece, 3, by the umber of times we would eed to add it to. 1 Elemets of these problems were adapted from Itegrated Mathematics 3 by McDougal-Littell, 00.) April, 011 Page 8 of 34
This gives us the followig explicit formula for a arithmetic sequece: a a1 1 d. f. Write this versio of the explicit formula for the sequece i this problem. Show how this versio is equivalet to the versio above. g. Ca you come up with a reaso for which you would wat to add up the radii of the cocetric circles that make up the target (for the purpose of the cotest)? Explai. h. Plot the sequece from this problem o a coordiate grid. What should you use for the idepedet variable? For the depedet variable? What type of graph is this? How does the a equatio of the recursive formula relate to the graph? How does the parameter d i the explicit form relate to the graph? i. Describe (usig y-itercept ad slope), but do ot graph, the plots of the arithmetic sequeces defied explicitly or recursively as follows: 1. 4 a 3 1 3. a 3 4.5 3. 1 a1 a1 10. 1 4. a a 1 a a 1 5. The Area of the Target: To decide o prizes for the archery cotest, your group decided to use the areas of the ceter ad rigs. You decided that rigs with smaller areas should be worth more poits. But how much more? Complete the followig ivestigatio to help you decide. a. Fid the sequece of the areas of the rigs, icludig the ceter. (Be careful.) b. Write a recursive formula ad a explicit formula for this sequece. c. If the target was larger, what would be the area of the 5 th rig? d. Fid the total area of the bull s eye by addig up the areas i the sequece. April, 011 Page 9 of 34
e. Cosider the followig sum: S a1 a a3... a 1 a a. Explai why that equatio is equivalet to S a1 a1 d a1 d... a d a d a. Rewrite this latter equatio ad the write it out backwards. Add the two resultig equatios. Use this to fiish derivig the formula for the sum of the terms i a arithmetic sequece. Try it out o a few differet short sequeces. f. Use the formula for the sum of a fiite arithmetic sequece i part (e) to verify the sum of the areas i the target from part (d). g. Sometimes, we do ot have all the terms of the sequece but we still wat to fid a specific sum. For example, we might wat to fid the sum of the first 15 multiples of 4. Write a explicit formula that would represet this sequece. Is this a arithmetic sequece? If so, how could we use what we kow about arithmetic sequeces ad the sum formula i (e) to fid this sum? Fid the sum. h. What happes to the sum of the arithmetic series we ve bee lookig at as the umber of terms we sum gets larger? How could you fid the sum of the first 00 multiples of 4? How could you fid the sum of all the multiples of 4? Explai usig a graph ad usig mathematical reasoig. i. Let s practice a few arithmetic sum problems. 1. Fid the sum of the first 50 terms of 15, 9, 3, -3,. Fid the sum of the first 100 atural umbers 3. Fid the sum of the first 75 positive eve umbers 4. Come up with your ow arithmetic sequece ad challege a classmate to fid the sum. j. Summarize what you leared / reviewed about arithmetic sequeces ad series durig this task. April, 011 Page 10 of 34
3. Poit Values: Assume that each participat s arrow hits the surface of the target. a. Determie the probability of hittig each rig ad the bull s-eye. Target Piece Area of Piece (i cm ) Probability of Hittig this Area Bull s Eye 16 Rig 1 48 Rig 80 Rig 3 11 Rig 4 144 Rig 5 176 Rig 6 08 Rig 7 40 Rig 8 7 Rig 9 304 b. Assig poit values for hittig each part of the target, justifyig the amouts based o the probabilities just determied. c. Use your aswer to (b) to determie the expected umber of poits oe would receive after shootig a sigle arrow. d. Usig your aswers to part (c), determie how much you should charge for participatig i the cotest OR for what poit values participats would wi a prize. Justify your decisios. April, 011 Page 11 of 34
Group Two: Rock Throwig Cotest For the rock throwig cotest, your group decided to provide three differet arragemets of cas for participats to kock dow. 1. For the first arragemet, the ti cas were set up i a triagular patter, oly oe ca deep. (See picture.) a. If the top row is cosidered to be row 1, how may cas would be o row 10? b. Is this a arithmetic or a geometric sequece (or either)? Write explicit ad recursive formulas for the sequece that describes the umber of cas i the th row of this arragemet. c. It is importat to have eough cas to use i the cotest, so your group eeds to determie how may cas are eeded to make this arragemet. Make a table of the umber of rows icluded ad the total umber of cas. Rows Icluded Total Cas 1 1 3 3 4 5 6 7 8 d. Oe of your group members decides that it would be fu to have a mega-pyramid 0 rows high. You eed to determie how may cas would be eeded for this pyramid, but you do t wat to add all the umbers together. Oe way to fid the sum is to use the summatio formula you foud i the Archery Cotest. How do you fid the sum i a arithmetic sequece? Fid the sum of a pyramid arragemet 0 rows high usig this formula. Adapted from Maouchehri, A. (007). Iquiry-discourse mathematics istructio. Mathematics Teacher, 101, 90 300. April, 011 Page 1 of 34
We ca also write this problem usig summatio otatio: i 1 a, where i is the idex of summatio, is the upper limit of summatio, ad 1 is the lower limit of summatio. We ca thik of a i as the explicit formula for the sequece. I this pyramid problem, we have 0 i i 1 because we are summig the umbers from 1 to 0. We also kow what this sum is equal to. 0 0 i a1 a 1 0. What if we did ot kow the value of, the upper limit but we i 1 did kow that the first umber is 1 ad that we were coutig up by 1s? We would the have 1 i 1. This is a very commo, importat formula i sequeces. We will use i 1 it agai later. e. Propose ad justify a specific umber of cas that could be used i this triagular arragemet. Remember, it must be realistic for your fellow studets to stad or sit ad throw a rock to kock dow the cas. It must also be reasoable that the cas could be set back up rather quickly. Cosider restrictig yourself to less tha 50 cas for each pyramid. Describe the set-up ad exactly how may cas you eed. i. For the secod arragemet, the group decided to make aother triagular arragemet; however, this time, they decided to make the pyramid or 3 cas deep. (The picture shows the -deep arragemet.) a. This arragemet is quite similar to the first arragemet. Write a explicit formula for the sequece describig the umber of cas i the th row if there are cas i the top row, as pictured. b. Determie the umber of cas eeded for the 0 th row. c. Similar to above, we eed to kow how may cas are eeded for this arragemet. How will this sum be related to the sum you foud i problem 1? d. The formula give above i summatio otatio oly applies whe we are coutig by oes. What are we coutig by to determie the umber of cas i each row? What if the cas were three deep? What would we be coutig by? I this latter case, how would the sum of the cas eeded be related to the sum of the cas eeded i the arragemet i problem 1? April, 011 Page 13 of 34
e. This leads us to a extremely importat property of sums: costat. What does this property mea? Why is it useful? ca c a, where c is a i i 1 i 1 i f. Suppose you wated to make a arragemet that is 8 rows high ad 4 cas deep. Use the property i e to help you determie the umber of cas you would eed for this arragemet. g. Propose ad justify a specific umber of cas that could be used i this triagular arragemet. You may decide how may cas deep (>1) to make the pyramid. Cosider restrictig yourself to less tha 50 cas for each pyramid. Describe the set-up ad exactly how may cas you eed. Show ay calculatios. 3. For the third arragemet, you had the idea to make the pyramid of cas resemble a true pyramid. The model you proposed to the group had 9 cas o bottom, 4 cas o the secod row, ad 1 ca o the top row. a. Complete the followig table. Row Number of Cas Chage from Previous Row 1 1 1 4 3 3 9 5 4 5 6 7 8 9 10 b. How may cas are eeded for the th row of this arragemet? April, 011 Page 14 of 34
c. What do you otice about the umbers i the third colum above? Write a equatio that relates colum two to colum three. The try to write the equatio usig summatio otatio. d. How could you prove the relatioship you idetified i 3c? e. Let s look at a couple of ways to prove this relatioship. Cosider a visual approach to a proof. 3 Explai how you could use this approach to prove the relatioship. 1 f. We kow that the sum of the first atural umber is, so 1 1 3.... If we multiply both sides of the equatio by, we get the sum of the first EVEN umbers. How ca we use this ew equatio to help us fid the sum of the first ODD umbers? g. Cosider aother approach. We have S 1 3 5... 1. If we reverse the orderig i this equatio, we get S 1... 5 3 1. What happes if we add the correspodig terms of these two equatios? How will that help us prove the relatioship we foud earlier? h. I a future task, you will lear aother way to prove this relatioship. You will also look at the sum of rows i this ca arragemet. Ca you cojecture a formula for the sum of the first square umbers? Try it out a few times. 3 A similar approach ca be foud i the August 006 issue of the Mathematics Teacher: Activities for Studets: Visualizig Summatio Formulas by Guha Caglaya. April, 011 Page 15 of 34
4. Throughout this task, you leared a umber of facts ad properties about summatio otatio ad sums. You leared what summatio otatio is ad how to compute some sums usig the otatio. There is aother importat property to lear that is helpful i computig sums. We ll look at that here, alog with practicig usig summatio otatio. a. Write out the terms of these series. i. ii. iii. 4 i 1 5 i 1 5 i 1 5i i i 6 3 b. You have already see oe sum property. Here are the importat properties you eed to kow. Explai why the two ew properties make mathematical sese to you. Properties of sums (c represets a costat) 1.. 3. ca c a i i 1 i 1 c c i 1 i i i i i 1 i 1 i 1 i a b a b c. Express each series usig summatio otatio. The fid the sum. i. + 4 + 6 + + 4 ii. 5 + 8 + 11 + 14 + + 41 d. Compute each sum usig the properties of sums. i. ii. iii. 0 i 1 0 i 1 0 i 1 4i 3i 4 4i April, 011 Page 16 of 34
FASCINATING FRACTALS Sequeces ad series arise i may classical mathematics problems as well as i more recetly ivestigated mathematics, such as fractals. The task below ivestigates some of the iterestig patters that arise whe ivestigatig maipulatig differet figures. Part Oe: Koch Sowflake 4 (Images obtaied from Wikimedia Commos at http://commos.wikimedia.org/wiki/koch_sowflake) This shape is called a fractal. Fractals are geometric patters that are repeated at ever smaller icremets. The fractal i this problem is called the Koch sowflake. At each stage, the middle third of each side is replaced with a equilateral triagle. (See the diagram.) To better uderstad how this fractal is formed, let s create oe! O a large piece of paper, costruct a equilateral triagle with side legths of 9 iches. 4 Ofte the first picture is called stage 0. For this problem, it is called stage 1. The Sierpiski Triagle, the ext problem, presets the iitial picture as Stage 0. A umber of excellet applets are available o the web for viewig iteratios of fractals. April, 011 Page 17 of 34
Now, o each side, locate the middle third. (How may iches will this be?) Costruct a ew equilateral triagle i that spot ad erase the origial part of the triagle that ow forms the base of the ew, smaller equilateral triagle. How may sides are there to the sowflake at this poit? (Double-check with a parter before cotiuig.) Now cosider each of the sides of the sowflake. How log is each side? Locate the middle third of each of these sides. How log would oe-third of the side be? Costruct ew equilateral triagles at the middle of each of the sides. How may sides are there to the sowflake ow? Note that every side should be the same legth. Cotiue the process a few more times, if time permits. 1. Now complete the first three colums of the followig chart. Number of Legth of each Perimeter (i) Segmets Segmet (i) Stage 1 3 9 7 Stage Stage 3. Cosider the umber of segmets i the successive stages. a. Does the sequece of umber of segmets i each successive stage represet a arithmetic or a geometric sequece (or either)? Explai. b. What type of graph does this sequece produce? Make a plot of the stage umber ad umber of segmets i the figure to help you determie what type of fuctio you will use to model this situatio. c. Write a recursive ad explicit formula for the umber of segmets at each stage. d. Fid the 7 th term of the sequece. Fid 1 th term of the sequece. Now fid the 16 th. Do the umbers surprise you? Why or why ot? 3. Cosider the legth of each segmet i the successive stages. a. Does this sequece of legths represet a arithmetic or a geometric sequece (or either)? Explai. April, 011 Page 18 of 34
b. Write a recursive ad explicit formula for the legth of each segmet at each stage. c. Fid the 7 th term of the sequece. Fid the 1 th term of the sequece. Now fid the 16 th. How is what is happeig to these umbers similar or differet to what happeed to the sequece of the umber of segmets at each stage? Why are these similarities or differeces occurrig? 4. Cosider the perimeter of the Koch sowflake. a. How did you determie the perimeter for each of the stages i the table? b. Usig this idea ad your aswers i the last two problems, fid the approximate perimeters for the Koch sowflake at the 7 th, 1 th, ad 16 th stages. c. What do you otice about how the perimeter chages as the stage icreases? d. Extesio: B. B. Madelbrot used the ideas above, i.e. the legth of segmets ad the associated perimeters, i his discussio of fractal dimesio ad to aswer the questio, How log is the coast of Britai? Research Madelbrot s argumet ad explai why some might argue that the coast of Britai is ifiitely log. 5. Up to this poit, we have ot cosidered the area of the Koch sowflake. a. Usig whatever method you kow, determie the exact area of the origial triagle. b. How do you thik we might fid the area of the secod stage of the sowflake? What about the third stage? The 7 th stage? Are we addig area or subtractig area? c. To help us determie the area of the sowflake, complete the first two colums of the followig chart. Note: The sequece of the umber of ew triagles is represeted by a geometric sequece. Cosider how the umber of segmets might help you determie how may ew triagles are created at each stage. Stage Number of Segmets New triagles created Area of each of the ew triagles April, 011 Page 19 of 34 Total Area of the New Triagles 1 3 -- -- -- 3 4 5
d. Determie the exact areas of the ew triagles ad the total area added by their creatio for Stages 1 4. Fill i the chart above. (You may eed to refer back to problem 1 for the segmet legths.) Stage Number of Segmets New triagles created Area of each of the ew triagles Total Area of the New Triagles 1 3 -- -- -- 3 4 5 e. Because we are primarily iterested i the total area of the sowflake, let s look at the last colum of the table. The values form a sequece. Determie if it is arithmetic or geometric. The write the recursive ad explicit formulas for the total area added by the ew triagles at the th stage. f. Determie how much area would be added at the 10 th stage. 6. Rather tha lookig at the area at a specific stage, we are more iterested i the TOTAL area of the sowflake. So we eed to sum the areas. However, these are ot ecessarily umbers that we wat to try to add up. Istead, we ca use our rules of expoets ad properties of summatios to help us fid the sum. a. Write a expressio usig summatio otatio for the sum of the areas i the sowflake. b. Explai how the expressio you wrote i part a is equivalet to 4 81 3 9 4 3 4 4 4 3 9 i i. Now, the oly part left to determie is how to fid the sum of a fiite geometric series. Let s take a step back ad thik about how we form a fiite geometric series: S = a 1 + a 1 r + a 1 r + a 1 r 3 + + a 1 r -1 Multiplyig both sides by r, we get rs = a 1 r + a 1 r + a 1 r 3 ++ a 1 r 4 + a 1 r April, 011 Page 0 of 34
Subtractig these two equatios: S - rs = a 1 - a 1 r Factorig: S (1 r) = a 1 (1 r ) 1 Ad fially, 1 a r S 1 r c. Let s use this formula to fid the total area of oly the ew additios through the 5 th stage. (What is a 1 i this case? What is?) Check your aswer by summig the values i your table. d. Now, add i the area of the origial triagle. What is the total area of the Koch sowflake at the fifth stage? Do you thik it s possible to fid the area of the sowflake for a value of equal to ifiity? This is equivalet to fidig the sum of a ifiite geometric series. You ve already leared that we caot fid the sum of a ifiite arithmetic series, but what about a geometric oe? e. Let s look at a easier series: 1 + ½ + ¼ + Make a table of the first 10 sums of this series. What do you otice? Terms 1 3 4 5 6 7 8 9 10 Sum 1 3/ 7/4 f. Now, let s look at a similar series: 1 + + 4 +. Agai, make a table. How is this table similar or differet from the oe above? Why do thik this is so? Terms 1 3 4 5 6 7 8 9 10 Sum 1 3 7 Recall that ay real umber -1 < r < 1 gets smaller whe it is raised to a positive power; whereas umbers less tha -1 ad greater tha 1, i.e. r 1, get larger whe they are raised to a positive power. 1 Thikig back to our sum formula, 1 a r S, this meas that if r 1, as gets larger, r 1 r approaches 0. If we wat the sum of a ifiite geometric series, we ow have a1 1 0 a1 S. We say that if sum of a ifiite series exists i this case, the sum of a 1 r 1 r ifiite geometric series oly exists if r 1--the the series coverges to a sum. If a ifiite series does ot have a sum, we say that it diverges. All arithmetic series diverge. g. Of the series i parts e ad f, which would have a ifiite sum? Explai. Fid, usig the formula above, the sum of the ifiite geometric series. April, 011 Page 1 of 34
h. Write out the formula for the sum of the first terms of the sequece you summed i part g. Graph the correspodig fuctio. What do you otice about the graph ad the sum you foud?. i. Graphs ad ifiite series. Write each of the followig series usig sigma otatio. The fid the sum of the first 0 terms of the series; write out the formula. Fially, graph the fuctio correspodig to the sum formula for the first th terms. What do you otice about the umbers i the series, the fuctio, the sum, ad the graph? 3 1 1 1 1.... 3 3 3. 3 4 4 0.6 4 0.6 4 0.6... j. Let s retur to the area of the Koch Sowflake. If we cotiued the process of creatig ew triagles ifiitely, could we fid the area of the etire sowflake? Explai. k. If it is possible, fid the total area of the sowflake if the iteratios were carried out a ifiite umber of times. This problem is quite iterestig: We have a fiite area but a ifiite perimeter! April, 011 Page of 34
Part Two: The Sierpiski Triagle (Images take from Wikimedia Commos at http://e.wikipedia.org/wiki/file:sierpiski_triagle_evolutio.svg.) Aother example of a fractal is the Sierpiski triagle. Start with a triagle of side legth 1. This time, we will cosider the origial picture as Stage 0. I Stage 1, divide the triagle ito 4 cogruet triagles by coectig the midpoits of the sides, ad remove the ceter triagle. I Stage, repeat Stage 1 with the three remaiig triagles, removig the ceters i each case. This process repeats at each stage. 1. Mathematical Questios: Make a list of questios you have about this fractal, the Sierpiski triagle. What types of thigs might you wat to ivestigate?. Number of Triagles i the Evolutio of the Sierpiski Fractal a. How may shaded triagles are there at each stage of the evolutio? How may removed triagles are there? Use the table to help orgaize your aswers. Stage Number of Shaded Triagles 0 1 0 1 3 4 5 Number of Newly Removed Triagles b. Are the sequeces above the umber of shaded triagles ad the umber of ewly removed triagles arithmetic or geometric sequeces? How do you kow? c. How may shaded triagles would there be i the th stage? Write both the recursive ad explicit formulas for the umber of shaded triagles at the th stage. April, 011 Page 3 of 34
d. How may ewly removed triagles would there be i the th stage? Write both the recursive ad explicit formulas for the umber of ewly removed triagles at the th stage. e. We ca also fid out how may removed triagles are i each evolutio of the fractal. Write a expressio for the total umber of removed triagles at the th stage. Try a few examples to make sure that your expressio is correct. f. Fid the total umber of removed triagles at the 10 th stage. g. If we were to cotiue iteratig the Sierpiski triagle ifiitely, could we fid the total umber of removed triagles? Why or why ot. If it is possible, fid the sum. 3. Perimeters of the Triagles i the Sierpiski Fractal a. Assume that the sides i the origial triagle are oe uit log. Fid the perimeters of the shaded triagles. Complete the table below. Stage Legth of a Side of a Shaded Triagle Perimeter of each Shaded Triagle Number of Shaded Triagles 0 1 3 1 3 1 3 4 5 b. Fid the perimeter of the shaded triagles i the 10 th stage. Total Perimeter of the Shaded Triagles c. Is the sequece of values for the total perimeter arithmetic, geometric, or either? Explai how you kow. d. Write recursive ad explicit formulas for this sequece. I both forms, the commo ratio should be clear. 4. Areas i the Sierpiski Fractal a. Assume that the legth of the side of the origial triagle is 1. Determie the exact area of each shaded triagle at each stage. Use this to determie the total area of the shaded triagles at each stage. (Hit: How are the shaded triagles at stage alike or differet?) April, 011 Page 4 of 34
Stage Legth of a Side of a Shaded Triagle (i) Area of each Shaded Triagle (i ) Number of Shaded Triagles 0 1 1 1 3 3 4 5 Total Area of the Shaded Triagles (i ) b. Explai why both the sequece of the area of each shaded triagle ad the sequece of the total area of the shaded triagles are geometric sequeces. What is the commo ratio i each? Explai why the commo ratio makes sese i each case. c. Write the recursive ad explicit formulas for the sequece of the area of each shaded triagle. Make sure that the commo ratio is clear i each form. d. Write the recursive ad explicit formulas for the sequece of the total area of the shaded triagles at each stage. Make sure that the commo ratio is clear i each form. e. Propose oe way to fid the sum of the areas of the removed triagles usig the results above. Fid the sum of the areas of the removed triagles i stage 5. Aother way to fid the sum of the areas of the removed triagles is to fid the areas of the ewly removed triagles at each stage ad sum them. Use the followig table to help you orgaize your work. Stage Legth of a Side of a Removed Triagle (i) Area of each Newly Removed Triagle (i ) Number of Newly Removed Triagles Total Area of the Newly Removed Triagles (i ) 0 0 0 0 0 1 3 4 5 f. Write the explicit ad recursive formulas for the area of the removed triagles at stage. April, 011 Page 5 of 34
g. Write a expressio usig summatio otatio for the sum of the areas of the removed triagles at each stage. The use this formula to fid the sum of the areas of the removed triagles i stage 5. h. Fid the sum of the areas of the removed triagles at stage 0. What does this tell you about the area of the shaded triagles at stage 0? (Hit: What is the area of the origial triagle?) i. If we were to cotiue iteratig the fractal, would the sum of the areas of the removed triagles coverge or diverge? How do you kow? If it coverges, to what value does it coverge? Explai i at least two ways. April, 011 Page 6 of 34
Part Three: More with Geometric Sequeces ad Series Up to this poit, we have oly ivestigated geometric sequeces ad series with a positive commo ratio. We will look at some additioal sequeces ad series to better uderstad how the commo ratio impacts the terms of the sequece ad the sum. 1. For each of the followig sequeces, determie if the sequece is arithmetic, geometric, or either. If arithmetic, determie the commo differece d. If geometric, determie the commo ratio r. If either, explai why ot. a., 4, 6, 8, 10, d., 4, 8, 16, 3, b., -4, 6, -8, 10, e., -4, 8, -16, 3, c. -, -4, -6, -8, -10, f. -, -4, -8, -16, -3,. Ca the sigs of the terms of a arithmetic or geometric sequece alterate betwee positive ad egative? If ot, explai. If so, explai whe. 3. Write out the first 6 terms of the series otice? i 1 i, i 0 i, ad i 1 i 1 i 1. What do you 4. Write each series i summatio otatio ad fid sum of first 10 terms. a. 1 ½ + ¼ - 1/8 + b. 3 + ¾ + 3/16 + 3/64 + c. - 4 + 1 36 + 108-5. Which of the series above would coverge? Which would diverge? How do you kow? For the series that will coverge, fid the sum of the ifiite series. April, 011 Page 7 of 34
DIVING INTO DIVERSIONS Sequeces ad series ofte help us solve mathematical problems more efficietly tha without their help. Ofte, however, oe must make cojectures, test out the cojectures, ad the prove the statemets before the sequeces ad series ca be geeralized for specific situatios. This task explores some of the usefuless of sequeces ad series. So, let s dive ito some mathematical diversios o sequeces ad series! Part Oe: Fiboacci Sequece 5 1. Hoeybees ad Family Trees The hoeybee is a iterestig isect. Hoeybees live i coloies called hives ad they have uusual family trees. Each hive has a special female called the quee. The other females are worker bees; these females produce o eggs ad therefore do ot reproduce. The males are called droes. Here is what is really uusual about hoeybees: Male hoeybees hatch from ufertilized eggs laid by the quee ad therefore have a female paret but o male paret. Female hoeybees hatch from the quee s fertilized eggs. Therefore, males have oly 1 paret, a mother, whereas females have both a mother ad a father. Cosider the family tree of a male hoeybee. 5 The traditioal problem for examiig the Fiboacci sequece is the rabbit problem. The hoeybee problem, however, is more realistic. A umber of additioal Fiboacci activities are available o Dr. Kott s webpage at www.mcs.surrey.ac.uk/persoal/r.kott/fiboacci/fibpuzzles.html Lessos o the Rabbit problem ca be foud o Dr. Kott s page ad o NCTM Illumiatios. April, 011 Page 8 of 34
(Picture obtaied from http://www.chabad.org/media/images/11/tile11614.jpg o the webpage www.chabad.org.) a. I the first geeratio, there is 1 male hoeybee. This male hoeybee has oly oe paret, a mother, at geeratio. The third geeratio cosists of the male hoeybee s gradparets, the mother ad father of his mother. How may great-gradparets ad great-great gradparets does the male hoeybee have? Explai why this makes sese. b. How may acestors does the male hoeybee have at each previous geeratio? Complete the followig table. Fid a way to determie the umber of acestors without drawig out or coutig all the hoeybees. Term 1 3 4 5 6 7 8 9 10 # Value 1 1 3 5 8 13 1 34 55 c. The sequece of the umber of bees i each geeratio is kow as the Fiboacci sequece. Write a recursive formula for the Fiboacci sequece. (Hit: The recursio formula ivolves two previous terms.) F 1 = F = F =, April, 011 Page 9 of 34
d. The Fiboacci sequece is foud i a wide variety of objects that occur i ature: plats, mollusk shells, etc. Fid a picture of a item that exhibits the Fiboacci sequece ad explai how the sequece ca be see.. Golde Ratio The sequece of Fiboacci is ot the oly iterestig thig to arise from examiig the Hoeybee problem (or other similar pheomea). This ext problem ivestigates a amazig fact relatig the Fiboacci sequece to the golde ratio. a. Usig a spreadsheet or the list capabilities o your graphig calculator to create a list of the first 0 terms of the Fiboacci sequece ad their ratios. I the first colum, list the term umber. I the secod colum, record the value of that term. (Spreadsheets ca do this quickly.) I the third colum, make a list, begiig i the secod row, of each term ad its precedig term. For example, i the row with term, the calculate F /F 1. (Agai, a spreadsheet ca do this quite quickly. Record the ratios i the table below. What do you otice about the ratios? Term 3 4 5 6 7 8 9 10 11 Ratio Term 1 13 14 15 16 17 18 19 0 Ratio b. Create a plot of the term umber ad the ratio. What do you otice? April, 011 Page 30 of 34
c. Whe a sequece approaches a specific value i the log ru (as the term umber approaches ifiity), we say that the sequece has a limit. What appears to be the limit of the sequece of ratios of the terms i the Fiboacci sequece? d. It would be ice to kow the exact value of this limit. Take a step back. Fid the solutios to the quadratic equatio x x 1 = 0. Fid the decimal approximatio of the solutios. Does aythig look familiar? e. This umber is called the golde ratio ad is ofte writte as (phi). Just like the Fiboacci sequece, the golde ratio is preset i may aturally occurrig objects, icludig the proportios of our bodies. Fid a picture of a item that exhibits the golde ratio ad explai how the ratio is maifested. f. Retur to your spreadsheet ad create a ew colum of ratios. This time, fid the ratio of a term ad its followig term, e.g. F 3 /F 4. What appears to be the limit of this sequece of ratios? g. How do you thik the limit i part f might be related to the golde ratio? Test out your cojectures ad discuss with your classmates. April, 011 Page 31 of 34
Part Two: Does 0.9999 = 1? 1. I previous courses, you may have leared a variety of ways to express decimals as fractios. For termiatig decimals, this was relatively simple. The repeatig decimals, however, made for a much more iterestig problem. How might you chage 0.44444444. ito a fractio?. Oe way you may have expressed repeatig decimals as fractios is through the use of algebra. a. For istace, let x = 0.44444. What does 10x equal? You ow have two equatios. Subtract the two equatios ad solve for x. Here s your fractio! b. Use the algebra method to express each of the followig repeatig decimals as fractios. 0.7777. 0.454545. 0.55555 3. We ca also use geometric series to express repeatig decimals as fractios. Cosider 0.444444 agai. a. How ca we write this repeatig decimal as a ifiite series? b. Now fid the sum of this ifiite series. 4. Express each of the followig repeatig decimals as ifiite geometric series. The fid the sum of each ifiite series. a. 0.5 b. 0.47 c. 0.16 5. Let s retur to the origial questio: does 0.9999 = 1? Use at least two differet methods to aswer this questio. April, 011 Page 3 of 34
Part Three: Factorial Fu: Lots of useful sequeces ad series ivolve factorials. You first leared about factorials i Math 1 whe you discussed the biomial theorem, combiatios ad permutatios. 1. As a brief refresher, calculate each of the followig. Show how each ca be calculated without the use of a calculator. a. 0! b. 4! c. 5! 3! d. 10! 7!3!. Write out the first five terms of each of the followig sequeces. a. a b. a 1! c.! a 1 1 1! 3. Cosider the sequeces i umber. Do you thik each will diverge or coverge? Explai. (Hit: It may be helpful to plot the sequeces i a graphig calculator usig Seq mode.) 4. Write a explicit formula, usig factorials, for each of the followig sequeces. 1 1 1 a., 4, 1, 40, 40, b. 1,1,,,,... c. 1, 1, 3, 1,... 6 4 3 10 5 5. The Excitig Natural Base e I GPS Geometry, we saw that we could approximate the trascedetal umber e by cosiderig the compoud iterest formula. a. What is the formula for compouded iterest ad what value of e does your calculator provide? b. Fid the value of a $1 ivestmet at 100% for 1 year at differet values of. N 1 4 1 5 100 1000 10,000 100,000 Value of Ivestmet April, 011 Page 33 of 34
c. What ivestmet value is beig approached as icreased? So we say that as the value of icreased towards ifiity, the expressio approaches. d. We ca also approximate e usig a series with factorials. Write out ad sum the first te 1 terms of. What do you otice? i! i 1 e. Which approximatio, the oe employig the compoud iterest formula or the oe usig factorials, is more accurate with small values of? I the future, perhaps i calculus, you will cotiue ecouterig sequeces ad values that ca easily be expressed with factorials. Good luck! April, 011 Page 34 of 34