ࡱ> PRMNO bjbjcTcT 0>>P "81t\*T"((((Y)fGfGfG~ tfGFNfGfGfG(Y)4Y4Y4YfG(Y)4YfG4Y4Yt@(hOc L$E^0*JDP4YfGfGfGTBfGfGfG*fGfGfGfGfGfGfGfGfGfGfGfGfG :  Accelerated Mathematics III Frameworks Student Edition Unit 2 Sequences and Series  1st Edition June, 2010 Georgia Department of Education Table of Contents  TOC \o "1-3" \h \z \u  HYPERLINK \l "_Toc200527660" INTRODUCTION: 3  HYPERLINK \l "_Toc200527661" Renaissance Festival Learning Task 8  HYPERLINK \l "_Toc200527662" Fascinating Fractals Learning Task 17  HYPERLINK \l "_Toc200527663" Diving into Diversions Learning Task 28  Accelerated Mathematics III Unit 2 Sequences and Series Teachers Edition Introduction: In 8th and 9th grades, students learned about arithmetic and geometric sequences and their relationships to linear and exponential functions, respectively. This unit builds on students understandings of those sequences and extends students knowledge to include arithmetic and geometric series, both finite and infinite. Summation notation and properties of sums are also introduced. Additionally, students will examine other types of sequences and, if appropriate, proof by induction. They will use their knowledge of the characteristics of the types of sequences and the corresponding functions to compare scenarios involving different sequences. Enduring Understandings: All arithmetic and geometric sequences can be expressed recursively and explicitly. Some other sequences also can be expressed in both ways but others cannot. Arithmetic sequences are identifiable by a common difference and can be modeled by linear functions. Infinite arithmetic series always diverge. Geometric sequences are identifiable by a common ratio and can be modeled by exponential functions. Infinite geometric series diverge if  EMBED Equation.DSMT4  and converge is  EMBED Equation.DSMT4 . The sums of finite arithmetic and geometric series can be computed with easily derivable formulas. Identifiable sequences and series are found in many naturally occurring objects. Repeating decimals can be expressed as fractions by summing appropriate infinite geometric series. The principle of mathematical induction is a method for proving that a statement is true for all positive integers (or all positive integers greater than a specified integer). Key Standards Addressed: MA3A9. Students will use sequences and series a. Use and find recursive and explicit formulae for the terms of sequences. b. Recognize and use simple arithmetic and geometric sequences. c. Investigate limits of sequences. d. Use mathematical induction to find and prove formulae for sums of finite series. e. Find and apply the sums of finite and, where appropriate, infinite arithmetic and geometric series. f. Use summation notation to explore series. g. Determine geometric series and their limits. Related Standards Addressed: MA3A1. Students will explore rational function. Investigate and explore characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior. MA3A4. Students will investigate functions. Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise. Investigate transformations of functions. MA3P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. MA3P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. MA3P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely. MA3P4. Students will make connections among mathematical ideas and to other disciplines. a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics. MA3P5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical phenomena. unit overview: The launching activity begins by revisiting ideas of arithmetic sequences studied in eighth and ninth grades. Definitions, as well as the explicit and recursive forms of arithmetic sequences are reviewed. The task then introduces summations, including notation and operations with summations, and summing arithmetic series. The second set of tasks reviews geometric sequences and investigates sums, including infinite and finite geometric series, in the context of exploring fractals. It is assumed that students have some level of familiarity with geometric sequences and the relationship between geometric sequences and exponential functions. The third group addresses some common sequences and series, including the Fibonacci sequence, sequences with factorials, and repeating decimals. Additionally, mathematical induction is employed to prove that the explicit forms are valid. (Note: Induction is a standard for Accelerated Mathematics but could be used as enrichment with all students. However, other ways of proof are also addressed in the Renaissance Festival task: Rock Throwing Contest, problem 3.) Vocabulary and formulas Arithmetic sequence: A sequence of terms  EMBED Equation.DSMT4  with  EMBED Equation.DSMT4 . The explicit formula is given by  EMBED Equation.DSMT4  and the recursive form is a1 = value of the first term and  EMBED Equation.DSMT4 . Arithmetic series: The sum of a set of terms in arithmetic progression  EMBED Equation.DSMT4 with  EMBED Equation.DSMT4 . Common difference: In an arithmetic sequence or series, the difference between two consecutive terms is d,  EMBED Equation.DSMT4 . Common ratio: In a geometric sequence or series, the ratio between two consecutive terms is r,  EMBED Equation.DSMT4 . Explicit formula: A formula for a sequence that gives a direct method for determining the nth term of the sequence. It presents the relationship between two quantities, i.e. the term number and the value of the term. Factorial: If n is a positive integer, the notation n! (read n factorial) is the product of all positive integers from n down through 1; that is,  EMBED Equation.DSMT4 . Note that 0!, by definition, is 1; i.e. EMBED Equation.DSMT4 . Finite series: A series consisting of a finite, or limited, number of terms. Infinite series: A series consisting of an infinite number of terms. Geometric sequence: A sequence of terms  EMBED Equation.DSMT4  with  EMBED Equation.DSMT4 . The explicit formula is given by  EMBED Equation.DSMT4  and the recursive form is  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Geometric series: The sum of a set of terms in geometric progression  EMBED Equation.DSMT4 with  EMBED Equation.DSMT4 . Limit of a sequence: The long-run value that the terms of a convergent sequence approach. Partial sum: The sum of a finite number of terms of an infinite series. Recursive formula: Formula for determining the terms of a sequence. In this type of formula, each term is dependent on the term or terms immediately before the term of interest. The recursive formula must specific at least one term preceding the general term. Sequence: A sequence is an ordered list of numbers. Summation or sigma notation:  EMBED Equation.DSMT4 , where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation. This expression gives the partial sum, the sum of the first n terms of a sequence. More generally, we can write  EMBED Equation.DSMT4 , where k is the starting value. Sum of a finite arithmetic series: The sum, Sn, of the first n terms of an arithmetic sequence is given by  EMBED Equation.DSMT4 , where a1 = value of the first term and an = value of the last term in the sequence. Sum of a finite geometric series: The sum, Sn, of the first n terms of a geometric sequence is given by EMBED Equation.DSMT4 , where a1 is the first term and r is the common ratio (r ( 1). Sum of an infinite geometric series: The general formula for the sum S of an infinite geometric series  EMBED Equation.DSMT4  with common ratio r where  EMBED Equation.DSMT4  is  EMBED Equation.DSMT4 . If an infinite geometric series has a sum, i.e. if  EMBED Equation.DSMT4 , then the series is called a convergent geometric series. All other geometric (and arithmetic) series are divergent. Term of a sequence: Each number in a sequence is a term of the sequence. The first term is generally noted as a1, the second as a2, , the nth term is noted as an. an is also referred to as the general term of a sequence. Renaissance Festival Learning task As part of a class project on the Renaissance, your class decided to plan a renaissance festival for the community. Specifically, you are a member of different groups in charge of planning two of the contests. You must help plan the archery and rock throwing contests. The following activities will guide you through the planning process. Group One: Archery Contest Before planning the archery contest, your group decided to investigate the characteristics of the target. The target being used has a center, or bulls-eye, with a radius of 4 cm, and nine rings that are each 4 cm wide. 1. The Target Sketch a picture of the center and first 3 rings of the target. Write a sequence that gives the radius of each of the concentric circles that comprise the entire target. Write a recursive formula and an explicit formula for the terms of this sequence. What would be the radius of the target if it had 25 rings? Show how you completed this problem using the explicit formula. In the past, you have studied both arithmetic and geometric sequences. What is the difference between these two types of sequences? Is the sequence in (b) arithmetic, geometric, or neither? Explain. One version of the explicit formula uses the first term, the common difference, and the number of terms in the sequence. For example, if we have the arithmetic sequence 2, 5, 8, 11, 14, , we see that the common difference is 3. If we want to know the value of the 20th term, or a20, we could think of starting with a1 = 2 and adding the difference, d = 3 a certain number of times. How many times would we need to add the common difference to get to the 20th term? _____ Because multiplication is repeated addition, instead of adding 3 that number of times, we could multiply the common difference, 3, by the number of times we would need to add it to 2. This gives us the following explicit formula for an arithmetic sequence:  EMBED Equation.DSMT4 . Write this version of the explicit formula for the sequence in this problem. Show how this version is equivalent to the version above. Can you come up with a reason for which you would want to add up the radii of the concentric circles that make up the target (for the purpose of the contest)? Explain. Plot the sequence from this problem on a coordinate grid. What should you use for the independent variable? For the dependent variable? What type of graph is this? How does the an equation of the recursive formula relate to the graph? How does the parameter d in the explicit form relate to the graph? Describe (using y-intercept and slope), but do not graph, the plots of the arithmetic sequences defined explicitly or recursively as follows:  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  2. The Area of the Target: To decide on prizes for the archery contest, your group decided to use the areas of the center and rings. You decided that rings with smaller areas should be worth more points. But how much more? Complete the following investigation to help you decide. Find the sequence of the areas of the rings, including the center. (Be careful.) Write a recursive formula and an explicit formula for this sequence. If the target was larger, what would be the area of the 25th ring? Find the total area of the bulls eye by adding up the areas in the sequence. Consider the following sum:  EMBED Equation.DSMT4 . Explain why that equation is equivalent to  EMBED Equation.DSMT4 . Rewrite this latter equation and then write it out backwards. Add the two resulting equations. Use this to finish deriving the formula for the sum of the terms in an arithmetic sequence. Try it out on a few different short sequences. Use the formula for the sum of a finite arithmetic sequence in part (e) to verify the sum of the areas in the target from part (d). Sometimes, we do not have all the terms of the sequence but we still want to find a specific sum. For example, we might want to find the sum of the first 15 multiples of 4. Write an explicit formula that would represent this sequence. Is this an arithmetic sequence? If so, how could we use what we know about arithmetic sequences and the sum formula in (e) to find this sum? Find the sum. What happens to the sum of the arithmetic series weve been looking at as the number of terms we sum gets larger? How could you find the sum of the first 200 multiples of 4? How could you find the sum of all the multiples of 4? Explain using a graph and using mathematical reasoning. Lets practice a few arithmetic sum problems. Find the sum of the first 50 terms of 15, 9, 3, -3, Find the sum of the first 100 natural numbers Find the sum of the first 75 positive even numbers Come up with your own arithmetic sequence and challenge a classmate to find the sum. j. Summarize what you learned / reviewed about arithmetic sequences and series during this task. 3. Point Values: Assume that each participants arrow hits the surface of the target. a. Determine the probability of hitting each ring and the bulls-eye. Target PieceArea of Piece (in cm2)Probability of Hitting this AreaBulls Eye16(Ring 148(Ring 280(Ring 3112(Ring 4144(Ring 5176(Ring 6208(Ring 7240(Ring 8272(Ring 9304( b. Assign point values for hitting each part of the target, justifying the amounts based on the probabilities just determined. c. Use your answer to (b) to determine the expected number of points one would receive after shooting a single arrow. d. Using your answers to part (c), determine how much you should charge for participating in the contest OR for what point values participants would win a prize. Justify your decisions. Group Two: Rock Throwing Contest For the rock throwing contest, your group decided to provide three different arrangements of cans for participants to knock down. For the first arrangement, the tin cans were set up in a triangular pattern, only one can deep. (See picture.) If the top row is considered to be row 1, how many cans would be on row 10? Is this an arithmetic or a geometric sequence (or neither)? Write explicit and recursive formulas for the sequence that describes the number of cans in the nth row of this arrangement. It is important to have enough cans to use in the contest, so your group needs to determine how many cans are needed to make this arrangement. Make a table of the number of rows included and the total number of cans. Rows IncludedTotal Cans1123345678 One of your group members decides that it would be fun to have a mega-pyramid 20 rows high. You need to determine how many cans would be needed for this pyramid, but you dont want to add all the numbers together. One way to find the sum is to use the summation formula you found in the Archery Contest. How do you find the sum in an arithmetic sequence? ____________ Find the sum of a pyramid arrangement 20 rows high using this formula. We can also write this problem using summation notation:  EMBED Equation.DSMT4 , where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation. We can think of ai as the explicit formula for the sequence. In this pyramid problem, we have  EMBED Equation.DSMT4  because we are summing the numbers from 1 to 20. We also know what this sum is equal to.  EMBED Equation.DSMT4 . What if we did not know the value of n, the upper limit but we did know that the first number is 1 and that we were counting up by 1s? We would then have  EMBED Equation.DSMT4 . This is a very common, important formula in sequences. We will use it again later. Propose and justify a specific number of cans that could be used in this triangular arrangement. Remember, it must be realistic for your fellow students to stand or sit and throw a rock to knock down the cans. It must also be reasonable that the cans could be set back up rather quickly. Consider restricting yourself to less than 50 cans for each pyramid. Describe the set-up and exactly how many cans you need. For the second arrangement, the group decided to make another triangular arrangement; however, this time, they decided to make the pyramid 2 or 3 cans deep. (The picture shows the 2-deep arrangement.) This arrangement is quite similar to the first arrangement. Write an explicit formula for the sequence describing the number of cans in the nth row if there are 2 cans in the top row, as pictured. Determine the number of cans needed for the 20th row. Similar to above, we need to know how many cans are needed for this arrangement. How will this sum be related to the sum you found in problem 1? The formula given above in summation notation only applies when we are counting by ones. What are we counting by to determine the number of cans in each row? What if the cans were three deep? What would we be counting by? In this latter case, how would the sum of the cans needed be related to the sum of the cans needed in the arrangement in problem 1? This leads us to an extremely important property of sums: EMBED Equation.DSMT4 , where c is a constant. What does this property mean? Why is it useful?  Suppose you wanted to make an arrangement that is 8 rows high and 4 cans deep. Use the property in 2e to help you determine the number of cans you would need for this arrangement. Propose and justify a specific number of cans that could be used in this triangular arrangement. You may decide how many cans deep (>1) to make the pyramid. Consider restricting yourself to less than 50 cans for each pyramid. Describe the set-up and exactly how many cans you need. Show any calculations. For the third arrangement, you had the idea to make the pyramid of cans resemble a true pyramid. The model you proposed to the group had 9 cans on bottom, 4 cans on the second row, and 1 can on the top row. Complete the following table. RowNumber of CansChange from Previous Row11124339545678910 How many cans are needed for the nth row of this arrangement? What do you notice about the numbers in the third column above? Write an equation that relates column two to column three. Then try to write the equation using summation notation. How could you prove the relationship you identified in 3c? Lets look at a couple of ways to prove this relationship. Consider a visual approach to a proof. Explain how you could use this approach to prove the relationship.  We know that the sum of the first n natural number is  EMBED Equation.DSMT4 , so  EMBED Equation.DSMT4 . If we multiply both sides of the equation by 2, we get the sum of the first n EVEN numbers. How can we use this new equation to help us find the sum of the first n ODD numbers? Consider another approach. We have  EMBED Equation.DSMT4 . If we reverse the ordering in this equation, we get  EMBED Equation.DSMT4 . What happens if we add the corresponding terms of these two equations? How will that help us prove the relationship we found earlier? In a future task, you will learn another way to prove this relationship. You will also look at the sum of rows in this can arrangement. Can you conjecture a formula for the sum of the first n square numbers? Try it out a few times. Throughout this task, you learned a number of facts and properties about summation notation and sums. You learned what summation notation is and how to compute some sums using the notation. There is another important property to learn that is helpful in computing sums. Well look at that here, along with practicing using summation notation. Write out the terms of these series.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  You have already seen one sum property. Here are the important properties you need to know. Explain why the two new properties make mathematical sense to you. Properties of sums (c represents a constant) 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  Express each series using summation notation. Then find the sum. 2 + 4 + 6 + + 24 5 + 8 + 11 + 14 + + 41 Compute each sum using the properties of sums.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Fascinating Fractals lEARNING tASK: Sequences and series arise in many classical mathematics problems as well as in more recently investigated mathematics, such as fractals. The task below investigates some of the interesting patterns that arise when investigating manipulating different figures. Part One: Koch Snowflake       (Images obtained from Wikimedia Commons at  HYPERLINK "http://commons.wikimedia.org/wiki/Koch_snowflake" http://commons.wikimedia.org/wiki/Koch_snowflake) This shape is called a fractal. Fractals are geometric patterns that are repeated at ever smaller increments. The fractal in this problem is called the Koch snowflake. At each stage, the middle third of each side is replaced with an equilateral triangle. (See the diagram.) To better understand how this fractal is formed, lets create one! On a large piece of paper, construct an equilateral triangle with side lengths of 9 inches. Now, on each side, locate the middle third. (How many inches will this be?) Construct a new equilateral triangle in that spot and erase the original part of the triangle that now forms the base of the new, smaller equilateral triangle. How many sides are there to the snowflake at this point? (Double-check with a partner before continuing.) Now consider each of the sides of the snowflake. How long is each side? Locate the middle third of each of these sides. How long would one-third of the side be? Construct new equilateral triangles at the middle of each of the sides. How many sides are there to the snowflake now? Note that every side should be the same length. Continue the process a few more times, if time permits. 1. Now complete the first three columns of the following chart. Number of SegmentsLength of each Segment (in)Perimeter (in)Stage 13927Stage 2Stage 3 2. Consider the number of segments in the successive stages. Does the sequence of number of segments in each successive stage represent an arithmetic or a geometric sequence (or neither)? Explain. What type of graph does this sequence produce? Make a plot of the stage number and number of segments in the figure to help you determine what type of function you will use to model this situation. Write a recursive and explicit formula for the number of segments at each stage. d. Find the 7th term of the sequence. Find 12th term of the sequence. Now find the 16th. Do the numbers surprise you? Why or why not? 3. Consider the length of each segment in the successive stages. Does this sequence of lengths represent an arithmetic or a geometric sequence (or neither)? Explain. b. Write a recursive and explicit formula for the length of each segment at each stage. c. Find the 7th term of the sequence. Find the 12th term of the sequence. Now find the 16th. How is what is happening to these numbers similar or different to what happened to the sequence of the number of segments at each stage? Why are these similarities or differences occurring? 4. Consider the perimeter of the Koch snowflake. a. How did you determine the perimeter for each of the stages in the table? b. Using this idea and your answers in the last two problems, find the approximate perimeters for the Koch snowflake at the 7th, 12th, and 16th stages. c. What do you notice about how the perimeter changes as the stage increases? d. Extension: B. B. Mandelbrot used the ideas above, i.e. the length of segments and the associated perimeters, in his discussion of fractal dimension and to answer the question, How long is the coast of Britain? Research Mandelbrots argument and explain why some might argue that the coast of Britain is infinitely long. Up to this point, we have not considered the area of the Koch snowflake. Using whatever method you know, determine the exact area of the original triangle. How do you think we might find the area of the second stage of the snowflake? What about the third stage? The 7th stage? Are we adding area or subtracting area? To help us determine the area of the snowflake, complete the first two columns of the following chart. Note: The sequence of the number of new triangles is represented by a geometric sequence. Consider how the number of segments might help you determine how many new triangles are created at each stage. StageNumber of SegmentsNew triangles createdArea of each of the new trianglesTotal Area of the New Triangles13------23345n Determine the exact areas of the new triangles and the total area added by their creation for Stages 1 4. Fill in the chart above. (You may need to refer back to problem 1 for the segment lengths.) Because we are primarily interested in the total area of the snowflake, lets look at the last column of the table. The values form a sequence. Determine if it is arithmetic or geometric. Then write the recursive and explicit formulas for the total area added by the new triangles at the nth stage. Determine how much area would be added at the 10th stage. Rather than looking at the area at a specific stage, we are more interested in the TOTAL area of the snowflake. So we need to sum the areas. However, these are not necessarily numbers that we want to try to add up. Instead, we can use our rules of exponents and properties of summations to help us find the sum. Write an expression using summation notation for the sum of the areas in the snowflake. Explain how the expression you wrote in part a is equivalent to  EMBED Equation.DSMT4 . Now, the only part left to determine is how to find the sum of a finite geometric series. Lets take a step back and think about how we form a finite geometric series: Sn = a1 + a1r + a1r2 + a1r3 + + a1rn-1 Multiplying both sides by r, we get rSn = a1r + a1r2 + a1r3 ++ a1r4 + a1rn Subtracting these two equations: Sn - rSn = a1 - a1rn Factoring: Sn(1 r) = a1(1 rn) And finally,  EMBED Equation.DSMT4  Lets use this formula to find the total area of only the new additions through the 5th stage. (What is a1 in this case? What is n?) Check your answer by summing the values in your table. Now, add in the area of the original triangle. What is the total area of the Koch snowflake at the fifth stage? Do you think its possible to find the area of the snowflake for a value of n equal to infinity? This is equivalent to finding the sum of an infinite geometric series. Youve already learned that we cannot find the sum of an infinite arithmetic series, but what about a geometric one? Lets look at an easier series: 1 + + + Make a table of the first 10 sums of this series. What do you notice? Terms12345678910Sum13/27/4 Now, lets look at a similar series: 1 + 2 + 4 + . Again, make a table. How is this table similar or different from the one above? Why do think this is so? Terms12345678910Sum137 Recall that any real number -1 < r < 1 gets smaller when it is raised to a positive power; whereas numbers less than -1 and greater than 1, i.e.  EMBED Equation.DSMT4 , get larger when they are raised to a positive power. Thinking back to our sum formula,  EMBED Equation.DSMT4 , this means that if  EMBED Equation.DSMT4 , as n gets larger, rn approaches 0. If we want the sum of an infinite geometric series, we now have  EMBED Equation.DSMT4 . We say that if sum of an infinite series existsin this case, the sum of an infinite geometric series only exists if  EMBED Equation.DSMT4 --then the series converges to a sum. If an infinite series does not have a sum, we say that it diverges. All arithmetic series diverge. Of the series in parts e and f, which would have an infinite sum? Explain. Find, using the formula above, the sum of the infinite geometric series. Write out the formula for the sum of the first n terms of the sequence you summed in part g. Graph the corresponding function. What do you notice about the graph and the sum you found? Graphs and infinite series. Write each of the following series using sigma notation. Then find the sum of the first 20 terms of the series; write out the formula. Finally, graph the function corresponding to the sum formula for the first nth terms. What do you notice about the numbers in the series, the function, the sum, and the graph? 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  Lets return to the area of the Koch Snowflake. If we continued the process of creating new triangles infinitely, could we find the area of the entire snowflake? Explain. If it is possible, find the total area of the snowflake if the iterations were carried out an infinite number of times. This problem is quite interesting: We have a finite area but an infinite perimeter! Extension. Instead of using a triangle with a specific starting length, i.e. 9 inches, generalize the formulas for the Koch snowflakeperimeter and areain terms of the side length s. One possible resource that may be of use is Larry Riddles web site:  HYPERLINK "http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm" http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm Part Two: The Sierpinski Triangle  (Images taken from Wikimedia Commons at  HYPERLINK "http://en.wikipedia.org/wiki/File:Sierpinski_triangle_evolution.svg" http://en.wikipedia.org/wiki/File:Sierpinski_triangle_evolution.svg.) Another example of a fractal is the Sierpinski triangle. Start with a triangle of side length 1. This time, we will consider the original picture as Stage 0. In Stage 1, divide the triangle into 4 congruent triangles by connecting the midpoints of the sides, and remove the center triangle. In Stage 2, repeat Stage 1 with the three remaining triangles, removing the centers in each case. This process repeats at each stage. 1. Mathematical Questions: Make a list of questions you have about this fractal, the Sierpinski triangle. What types of things might you want to investigate? 2. Number of Triangles in the Evolution of the Sierpinski Fractal a. How many shaded triangles are there at each stage of the evolution? How many removed triangles are there? Use the table to help organize your answers. StageNumber of Shaded TrianglesNumber of Newly Removed Triangles01012345 Are the sequences abovethe number of shaded triangles and the number of newly removed trianglesarithmetic or geometric sequences? How do you know? How many shaded triangles would there be in the nth stage? Write both the recursive and explicit formulas for the number of shaded triangles at the nth stage. How many newly removed triangles would there be in the nth stage? Write both the recursive and explicit formulas for the number of newly removed triangles at the nth stage. We can also find out how many removed triangles are in each evolution of the fractal. Write an expression for the total number of removed triangles at the nth stage. Try a few examples to make sure that your expression is correct. Find the total number of removed triangles at the 10th stage. If we were to continue iterating the Sierpinski triangle infinitely, could we find the total number of removed triangles? Why or why not. If it is possible, find the sum. 3. Perimeters of the Triangles in the Sierpinski Fractal a. Assume that the sides in the original triangle are one unit long. Find the perimeters of the shaded triangles. Complete the table below. StageLength of a Side of a Shaded TrianglePerimeter of each Shaded TriangleNumber of Shaded TrianglesTotal Perimeter of the Shaded Triangles01313132345n b. Find the perimeter of the shaded triangles in the 10th stage. Is the sequence of values for the total perimeter arithmetic, geometric, or neither? Explain how you know. Write recursive and explicit formulas for this sequence. In both forms, the common ratio should be clear. 4. Areas in the Sierpinski Fractal a. Assume that the length of the side of the original triangle is 1. Determine the exact area of each shaded triangle at each stage. Use this to determine the total area of the shaded triangles at each stage. (Hint: How are the shaded triangles at stage alike or different?) StageLength of a Side of a Shaded Triangle (in)Area of each Shaded Triangle (in2)Number of Shaded TrianglesTotal Area of the Shaded Triangles (in2)011132345n Explain why both the sequence of the area of each shaded triangle and the sequence of the total area of the shaded triangles are geometric sequences. What is the common ratio in each? Explain why the common ratio makes sense in each case. Write the recursive and explicit formulas for the sequence of the area of each shaded triangle. Make sure that the common ratio is clear in each form. Write the recursive and explicit formulas for the sequence of the total area of the shaded triangles at each stage. Make sure that the common ratio is clear in each form. Propose one way to find the sum of the areas of the removed triangles using the results above. Find the sum of the areas of the removed triangles in stage 5. Another way to find the sum of the areas of the removed triangles is to find the areas of the newly removed triangles at each stage and sum them. Use the following table to help you organize your work. StageLength of a Side of a Removed Triangle (in)Area of each Newly Removed Triangle (in2)Number of Newly Removed TrianglesTotal Area of the Newly Removed Triangles (in2)0000012345n Write the explicit and recursive formulas for the area of the removed triangles at stage n. Write an expression using summation notation for the sum of the areas of the removed triangles at each stage. Then use this formula to find the sum of the areas of the removed triangles in stage 5. Find the sum of the areas of the removed triangles at stage 20. What does this tell you about the area of the shaded triangles at stage 20? (Hint: What is the area of the original triangle?) If we were to continue iterating the fractal, would the sum of the areas of the removed triangles converge or diverge? How do you know? If it converges, to what value does it converge? Explain in at least two ways. Part Three: More with Geometric Sequences and Series Up to this point, we have only investigated geometric sequences and series with a positive common ratio. We will look at some additional sequences and series to better understand how the common ratio impacts the terms of the sequence and the sum. 1. For each of the following sequences, determine if the sequence is arithmetic, geometric, or neither. If arithmetic, determine the common difference d. If geometric, determine the common ratio r. If neither, explain why not. a. 2, 4, 6, 8, 10, d. 2, 4, 8, 16, 32, b. 2, -4, 6, -8, 10, e. 2, -4, 8, -16, 32, c. -2, -4, -6, -8, -10, f. -2, -4, -8, -16, -32, 2. Can the signs of the terms of an arithmetic or geometric sequence alternate between positive and negative? If not, explain. If so, explain when. 3. Write out the first 6 terms of the series  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4 . What do you notice? 4. Write each series in summation notation and find sum of first 10 terms. a. 1 + - 1/8 + b. 3 + + 3/16 + 3/64 + c. - 4 + 12 36 + 108 - 5. Which of the series above would converge? Which would diverge? How do you know? For the series that will converge, find the sum of the infinite series. Diving into diversions lEARNING tASK: Sequences and series often help us solve mathematical problems more efficiently than without their help. Often, however, one must make conjectures, test out the conjectures, and then prove the statements before the sequences and series can be generalized for specific situations. This task explores some of the usefulness of sequences and series. So, lets dive into some mathematical diversions on sequences and series! Part One: Fibonacci Sequence 1. Honeybees and Family Trees The honeybee is an interesting insect. Honeybees live in colonies called hives and they have unusual family trees. Each hive has a special female called the queen. The other females are worker bees; these females produce no eggs and therefore do not reproduce. The males are called drones. Here is what is really unusual about honeybees: Male honeybees hatch from unfertilized eggs laid by the queen and therefore have a female parent but no male parent. Female honeybees hatch from the queens fertilized eggs. Therefore, males have only 1 parent, a mother, whereas females have both a mother and a father. Consider the family tree of a male honeybee.  (Picture obtained from  HYPERLINK "http://www.chabad.org/media/images/112/TilE1126142.jpg" http://www.chabad.org/media/images/112/TilE1126142.jpg on the webpage  HYPERLINK "http://www.chabad.org" www.chabad.org.) In the first generation, there is 1 male honeybee. This male honeybee has only one parent, a mother, at generation 2. The third generation consists of the male honeybees grandparents, the mother and father of his mother. How many great-grandparents and great-great grandparents does the male honeybee have? Explain why this makes sense. How many ancestors does the male honeybee have at each previous generation? Complete the following table. Find a way to determine the number of ancestors without drawing out or counting all the honeybees. Term #12345678910Value112 The sequence of the number of bees in each generation is known as the Fibonacci sequence. Write a recursive formula for the Fibonacci sequence. (Hint: The recursion formula involves two previous terms.) F1 = ______ F2 = _______ Fn = ________ , n ( 2 d. The Fibonacci sequence is found in a wide variety of objects that occur in nature: plants, mollusk shells, etc. Find a picture of an item that exhibits the Fibonacci sequence and explain how the sequence can be seen. 2. Golden Ratio The sequence of Fibonacci is not the only interesting thing to arise from examining the Honeybee problem (or other similar phenomena). This next problem investigates an amazing fact relating the Fibonacci sequence to the golden ratio. a. Using a spreadsheet or the list capabilities on your graphing calculator to create a list of the first 20 terms of the Fibonacci sequence and their ratios. In the first column, list the term number. In the second column, record the value of that term. (Spreadsheets can do this quickly.) In the third column, make a list, beginning in the second row, of each term and its preceding term. For example, in the row with term 2, the calculate F2/F1. (Again, a spreadsheet can do this quite quickly. Record the ratios in the table below. What do you notice about the ratios? TermRatio234567891011 TermRatio121314151617181920 Create a plot of the term number and the ratio. What do you notice? When a sequence approaches a specific value in the long run (as the term number approaches infinity), we say that the sequence has a limit. What appears to be the limit of the sequence of ratios of the terms in the Fibonacci sequence? It would be nice to know the exact value of this limit. Take a step back. Find the solutions to the quadratic equation x2 x 1 = 0. Find the decimal approximation of the solutions. Does anything look familiar? This number is called the golden ratio and is often written as ( (phi). Just like the Fibonacci sequence, the golden ratio is present in many naturally occurring objects, including the proportions of our bodies. Find a picture of an item that exhibits the golden ratio and explain how the ratio is manifested. Return to your spreadsheet and create a new column of ratios. This time, find the ratio of a term and its following term, e.g. F3/F4. What appears to be the limit of this sequence of ratios? How do you think the limit in part f might be related to the golden ratio? Test out your conjectures and discuss with your classmates. Extension. We can continue exploring the Fibonacci sequence and the golden ratio to make a geometric sequence. Consider a new Fibonacci sequence with f1 = 1 and f2 = (. Then our sequence is 1, (, 1 + (, 1 + 2(, ________, ________, ________, But wait! 1 + ( = (2 because ( is a solution to x2 x 1 = 0 ( x2 = x + 1. So we can replace the third term above with (2. Now our sequence is 1, (, (2, What should the fourth term be so that we have a geometric sequence? Figure out how the next 3 terms can be expressed only in terms of (, thus illustrating that we can combine the Fibonacci sequence and the golden ratio to make a new geometric sequence! Part Two: Does 0.9999 = 1? In previous courses, you may have learned a variety of ways to express decimals as fractions. For terminating decimals, this was relatively simple. The repeating decimals, however, made for a much more interesting problem. How might you change 0.44444444. into a fraction? One way you may have expressed repeating decimals as fractions is through the use of algebra. For instance, let x = 0.44444. What does 10x equal? You now have two equations. Subtract the two equations and solve for x. Heres your fraction! Use the algebra method to express each of the following repeating decimals as fractions. 0.7777. 0.454545. 0.255555 We can also use geometric series to express repeating decimals as fractions. Consider 0.444444 again. How can we write this repeating decimal as an infinite series? Now find the sum of this infinite series. 4. Express each of the following repeating decimals as infinite geometric series. Then find the sum of each infinite series. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  5. Lets return to the original question: does 0.9999 = 1? Use at least two different methods to answer this question. Part Three: Factorial Fun: Lots of useful sequences and series involve factorials. You first learned about factorials in Math 1 when you discussed the binomial theorem, combinations and permutations. 1. As a brief refresher, calculate each of the following. Show how each can be calculated without the use of a calculator. a. 0! b. 4! c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  2. Write out the first five terms of each of the following sequences. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  3. Consider the sequences in number 2. Do you think each will diverge or converge? Explain. (Hint: It may be helpful to plot the sequences in a graphing calculator using Seq mode.) 4. Write an explicit formula, using factorials, for each of the following sequences. a. 2, 4, 12, 40, 240, b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  5. The Exciting Natural Base e In Math 2, we saw that we could approximate the transcendental number e by considering the compound interest formula. What is the formula for compounded interest and what value of e does your calculator provide? Find the value of a $1 investment at 100% for 1 year at different values of n. NValue of Investment1241252100100010,000100,000 What investment value is being approached as n increased? So we say that as the value of n increased towards infinity, the expression approaches _____. We can also approximate e using a series with factorials. Write out and sum the first ten terms of  EMBED Equation.DSMT4 . What do you notice? Which approximation, the one employing the compound interest formula or the one using factorials, is more accurate with small values of n? In the future, perhaps in calculus, you will continue encountering sequences and values that can easily be expressed with factorials. Good luck! Part Four: And They All Came Falling Down (The Principle of Mathematical Induction) 1. Domino Designs With your partner, create a pattern of dominos such that when you tap the first domino, all others fall, one at a time. Be prepared to share your design with the class. Now form two patterns of dominos that do not work, each for a different reason. Be prepared to share your designs with the class. Make a list of conditions that must be met in order for the dominos to all fall down. As a class, discuss the fewest conditions needed. List those conditions 2. What you have witnessed is called the Principle of Mathematical Induction. Write the statement of the principle. 3. The Principle of Mathematical Induction is a form of proof. It allows us to prove lots of the formulas we derived earlier in the unit as well as lots of other mathematical facts. Lets take a look at how we could use induction to prove that the sum of the first n natural numbers is  EMBED Equation.DSMT4 . That is,  EMBED Equation.DSMT4 . Step 1 (Base Case). Show that S1 is true. First, to show that the first condition holds, we need to show that the statement is true when n = 1. When n = 1, there is only the one term on the LHS (left hand side) of the equation. So we have  EMBED Equation.DSMT4 . Simplifying on the right, we obtain 1 = 1. So the statement is true for S1. In terms of the dominos analogy, step one is equivalent to saying that the first domino will fall if pushed. Step 2 (Inductive Step). Show that if Sk is true, then Sk+1 is also true.  What is Sk+1 in this problem? That is, how does the RHS appear if LHS is  EMBED Equation.DSMT4 ? __________________________________________ This will be our goal as we go through the induction proof. Back to the proof: Second, we assume that the statement is true for Sk. That is, we assume that  EMBED Equation.DSMT4  is true for some integer k. This is called the inductive hypothesis. In terms of dominos, this statement says that somewhere along the domino line, a domino falls. What we are trying to prove is that if this particular domino falls, so will the next one, and the next one, etc. Now for the fun part! Because we want to see what happens if we add the next integer, we add k + 1 to both sides of the inductive hypothesis.  EMBED Equation.DSMT4  Now, we work with the RHS to show that it is equivalent to the goal we identified at the above stop sign. Here we go:  EMBED Equation.DSMT4  Ah, were done! So by the Principle of Mathematical Induction we have proved that  EMBED Equation.DSMT4  for every positive integer n. C. This next example will be a little more formal. (Anything that is not part of the official proof will be marked with a stop sign.) Lets also look at how to prove that 2 is a factor of n2 + 3n for all positive integers n. What does it mean for 2 to be a factor of an expression? Step 1 (Base Case). Show that S1 is true. We need to show that 2 is a factor of 12 + 3(1). Simplifying, we confirm that 2 is a factor of 4: 4 = 2(2). The statement is true. This shows that S1 is true. Step 2 (Inductive Step). Show that if Sk is true, then Sk+1 is also true. What are Sk and Sk+1 in this problem? We assume that the statement is true for Sk. That is, 2 is a factor of k2 + 3k. We will use this to show that Sk+1 is true. We need to show that 2 is a factor of (k+1)2 + 3(k+1). (k+1)2 + 3(k+1) = k2 + 2k + 1 + 3k + 3 = k2 + 5k + 4. We wish to use the inductive hypothesis to help us see that 2 is a factor of this expression. Look at the inductive hypothesis. How might we use it to help us? k2 + 5k + 4 = (k2 + 3k) + (2k + 4) = (k2 + 3k) + 2(k + 2). By the inductive hypothesis, 2 is a factor of (k2 + 3k). Also, 2 is a factor of 2(k + 2). So 2 is a factor of the sum. Therefore 2 is a factor of (k+1)2 + 3(k+1). By the principle of mathematical induction, 2 is a factor of n2 + 3n for every positive integer n. c. As a last example, lets use an extended principle of mathematical induction to prove that n2 > 2n + 1 for n ( 3. In this case, Step 1 shows that the statement is true for the base case; here, that base case is n = 3. Step 2 is the same as the original version, although we assume that k ( 3. Our conclusion will be that the statement is true for all integers greater than or equal to 3 rather than true for all positive integers, as weve said in the last two examples. Prove: n2 > 2n + 1 for n ( 3. Step 1 (Base Case). Show that S3 is true. 32 > 2(3) + 1 32 > 7 is true. So the statement is true for the base case, n = 3. Step 2 (Inductive Step): Show that if Sk is true (k( 3), then Sk+1 is also true. What are Sk and Sk+1 in this problem? (k+1)2 > 2(k+1) + 1 k2 + 2k + 1 > 2k + 2 + 1 k2 + 1 > 3 k2 > 2. This statement is always true for k ( 3. So by the principle of mathematical induction, n2 > 2n + 1 for n ( 3. Use induction to prove each of the following.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  2 is a factor of n2 n for all positive integers n 2n > n2 for n ( 5. (Use the extended principle of induction)  Elements of these problems were adapted from Integrated Mathematics 3 by McDougal-Littell, 2002.)  Adapted from Manouchehri, A. (2007). Inquiry-discourse mathematics instruction. Mathematics Teacher, 101, 290300.  A similar approach can be found in the August 2006 issue of the Mathematics Teacher: Activities for Students: Visualizing Summation Formulas by Gunhan Caglayan.  Often the first picture is called stage 0. For this problem, it is called stage 1. The Sierpinski Triangle, the next problem, presents the initial picture as Stage 0. A number of excellent applets are available on the web for viewing iterations of fractals.  The traditional problem for examining the Fibonacci sequence is the rabbit problem. The honeybee problem, however, is more realistic. A number of additional Fibonacci activities are available on Dr. Knotts webpage at  HYPERLINK "http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpuzzles.html" www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpuzzles.html Lessons on the Rabbit problem can be found on Dr. Knotts page and on NCTM Illuminations.     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