Mathematical Problem Solving for Elementary School Teachers. Dennis E. White

Transcription

1 Mathematical Problem Solving for Elementary School Teachers Dennis E. White April 15, 2013

2 ii Copyright Copyright c 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, Dennis White, University of Minnesota. All rights reserved. This document or any portion thereof may not be reproduced without written permission of Dennis White. Any copy of this document or portion thereof must include a copy of this notice. Preface These notes were written over an ten year period in conjunction with the development of a mathematics course aimed at elementary education majors. The course had its inception in an ad hoc committee formed to address the Mathematical Association of America s document A Call for Change, which itself was a response to the National Council of Teachers of Mathematics Curriculum and Evaluation Standards. A Call for Change calls for a restructuring of how we teach and what we teach elementary education students. Certain fundamental principles guided the content and pedagogy of these notes. i. The mathematical content is nontrivial and nonremedial. We assume the students have basic manipulative skills. We do not teach remedial skills. We do not teach many topics in a superficial way. The problems are, for the most part, nontrivial. Some topics are explored to a depth often found in junior and senior level courses. There is little emphasis placed on drill exercises or memorization. ii. The general topics should conform to those described in A Call for Change. Topics include geometry, number theory, algebraic structures, analysis, probability and statistics. iii. As also mentioned in A Call for Change, special emphasis is given to the interconnection of ideas, to the communication of mathematics and to problem solving skills. Material in these notes interconnect in various ways. Many problems emphasize communicating mathematical ideas both orally and in writing. Many of the problems are open-ended. Some can be solved using a variety of techniques. iv. The course should be given in a non-threatening environment. It is intended that these notes be used in a cooperative learning environment. It is also intended that there would be less emphasis on tests and benchmarks. The experience the students using these notes have will be taken back to their own classrooms. To the Students These notes are substantially different from the mathematics textbooks you may be familiar with. There are no large bodies of exercises at the end of each

3 iii section. There are few drill exercises. There is little repetition. The text is densely written and requires close and careful reading. Later chapters frequently refer to results, exercises, and ideas from earlier chapters. However, these notes are meant to give you a greater understanding of (and maybe appreciation for) how mathematical problems are really solved. You will often be led through a series of exercises to an understanding of some fairly deep mathematical results. It is my belief that most students, with proper mathematical skills, can learn some fairly sophisticated mathematics. These notes will probably require more effort on your part than perhaps you have put into other mathematics courses. This is on purpose. I believe that learning mathematics takes active participation, including testing hypotheses, constructing examples, forming strategies, and organizing ideas. All these things you must do. The notes can t do them for you; your instructor can t do them for you. Learning mathematics is an active process. It is not possible to learn mathematics by reading a textbook like a novel. Good mathematics students, from elementary school to graduate school, read a math book with pencil and paper in hand. Mathematics is not a collection of independent topics. It is not Algebra I, Algebra II, Trigonometry, Plane Geometry, etc. All of mathematics is interconnected in a fundamental way. In these notes, you will find some problems which require methods from several different mathematical areas. Other problems have more than one solution, each solution coming from a different mathematical area. Ideas learned in Chapter 2 will reappear in Chapter 7, Chapter 11 and Chapter 12. Chapter 12 uses ideas from Chapter 2, Chapter 5, Chapter 6, and Chapter 10. The topics were chosen because they are related to material that is widely found in elementary school curricula. However, these topics are taught at a substantially deeper level. It is not the purpose of these notes to teach you elementary school mathematics. As I mentioned above, it is my belief that most students, with proper mathematical skills, can learn the material in these notes. However, students without those prerequisite skills have great difficulty. Those skills include facility at handling fractions, powers, exponents and radicals in both numeric and algebraic contexts. Students should also understand the basics of analytic geometry: graphing functions, linear and quadratic equations, the quadratic formula, etc. They should also be somewhat familiar with logarithms and with techniques of counting. Let me conclude with a word about calculators. For the most part, I have not discouraged the use of a calculator. Some exercises explicitly call for a calculator. However, besides the mathematical skills described in the preceding paragraph, another prerequisite is an understanding of appropriate calculator use. A calculator is not a substitute for mathematical common sense. It should not be used to divide 24 by 6, or to decide if 1 2 is larger or smaller than 1 3.

4 iv To the Instructors These notes were designed to be used in a problem-solving environment. I feel they work best in a cooperative group setting. They are not designed to be lectured from, and I don t think they work particularly well in that role. Nevertheless, lectures do have a place in this course, if they are short and appropriate. You will surely notice the paucity of drill exercises. While some mathematical topics do require drill exercises, my concentration in these notes is on problem-solving skills. If you feel your class needs extra drill exercises in a particular area, you are welcome to design your own. Because there are few exercises, I expect that most classes will do a sizable percentage of them. I leave it to your judgment which to omit. Many of the exercises are not routine. Many have more than one solution method. Consequently, your classroom role is much expanded over a typical lecture-style course. A few of the exercises have been starred. The star means two things. First, the exercise is difficult. Second, the exercise is not in the main flow of ideas, and can be omitted. The last chapter, Chapter 13, is an alternative to Chapter 8. Acknowledgments I would like to thank all the instructors and teaching assistants over the years who have helped me in the many revisions of these notes. I want to thank especially Bert Fristedt (who also provided Chapter 8) and Dennis Stanton for their many useful suggestions and comments.

5 Contents 1 Number Sequences Recursions Explicit Formulas Summing Arithmetic Sequences Summing Geometric Sequences Examples Fibonacci Numbers Tower of Hanoi Divisions of a plane Counting When to Add Permutations Combinations Selections with Repetitions Card Games Catalan Numbers Several Counting Problems One-to-One Correspondences The Recursion The Explicit Formula Graphs Eulerian Circuits Special Graphs Planar Graphs Polyhedra Tessellations Torus Graphs Coloring Graphs Tournaments v

6 vi CONTENTS 5 Integers and Rational Numbers Primes and Prime Factorization The Euclidean Algorithm and the GCD Number Bases Integers Rational Numbers Countability of the Rational Numbers Modular Arithmetic Examples Rules Solving Equations Divisibility Tests Nim Probability and Statistics, I Equiprobable The General Model Binomial Model Misuse of Statistics Graphs Conditional Probabilities Vector Geometry Review of some plane geometry Parametric representations of lines Distances and norms Orthogonality and perpendicularity Three-dimensional space Pythagorean triples Trees Counting Trees Minimum Spanning Trees Rooted Trees and Forests Real and Complex Numbers Irrational Numbers Rational Approximations The Intermediate Value Theorem Complex Numbers Zeros of Polynomials Algebraic Extensions and Zeros of Polynomials Infinities Constructible Numbers

7 CONTENTS vii 11 Probability and Statistics, II Expectation Central Measures Measures of the Spread The Central Limit Theorem Applying the Central Limit Theorem Odds Finite Fields A Review of Modular Arithmetic A Tournament A Field with Four Elements Constructing Finite Fields Tournaments Revisited Areas and Triangles Simple Areas Similar Figures Pythagorean theorem Pythagorean triples Circles Volumes

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9 List of Figures 1.1 Triangular numbers Triangular, square and pentagonal numbers Puff pastry Three fractal snowflakes Fractal snowflake Domino tilings of 2 1, 2 2, 2 3 and 2 4 rectangles Construction of f Construction of f Tower of Hanoi Dividing a plane Bill s block walk Triangulations of a quadrilateral Triangulations of a pentagon Triangulation of a hexagon Blockwalking Outlines with three headings Tableaux of children A 2 8 tableau Another 2 8 tableau Hint: count the shaded blocks in each row people, one handshake people, one handshake people, two handshakes Decomposition of a triangulation Another decomposition of a triangulation Partial triangulation of a 24-gon Another partial triangulation of a 24-gon Yet another partial triangulation of a 24-gon A blockwalk northwest of the tracks Decomposition of a blockwalk A 14-by-14 block-walking grid A 20-by-20 block-walking grid ix

10 x LIST OF FIGURES 3.22 The reflection principle Another reflected path Two bad paths A non-square block-walk grid Block-walk grid with a lake Length 4 block-walk Length 6 block walk Find non-crossing paths on a grid The bridges of Königsberg Königsberg graph A graph with labeled vertices and edges Which have Eulerian circuits? A graph with loops and multiple edges A non-connected Graph Finding an Eulerian circuit The handshake theorem Null graphs N 3 and N Complete graphs A bipartite graph K 2,2 and K 2, Three ways of drawing the same graph Two ways of drawing K Planar graphs? K 3,3 in the Peterson graph Three trees Two planar embeddings of the same graph A planar graph Another planar graph Degree of a face in a planar graph A cube Planar view of a tetrahedron A truncated tetrahedron A hexagonal prism A regular tessellation Not a regular tessellation A semi-regular tessellation A torus A flattened-out torus A planar map and its dual Coloring a degree 5 vertex A tournament Nim Game Continuation of Nim Game Continuation of Nim Game

11 LIST OF FIGURES xi 6.4 Another Nim Game Yet Another Nim Game Still Another Nim Game One tree diagram Another tree diagram First golf ball tree Second golf ball tree Conditional probability tree for genetics Second conditional probability tree for genetics Two congruent right triangles The Pythagorean Theorem via comparison of areas A line represented parametrically Three tree networks with three cities Two-city tree network Two tree networks with four cities A five-city tree network The reduced tree network Further reduced tree network Final reduced tree network Degrees of cities First edge Two edges Three edges All edges restored Towns with degrees First edge Second edge Third edge Fourth edge Final tree network Another tree network Still another tree network Yet another tree network Another tree network Snowtown, Minnesota Streets plowed in Snowtown, Minnesota A graph with costs Another graph with costs K 4 with costs Another K 4 with costs Rooted tree Tree of mushrooms Same tree?

12 xii LIST OF FIGURES 9.32 Planar rooted trees A planar forest A different planar forest Labeled rooted tree A labeled forest Superbowl tournament Binary trees A counting problem Another counting problem A convex set and a non-convex set Slicing a convex set Construction of Cosine of α = a/b Construction of the Cosine Fifty Dice Rolls Normal Distribution Area Under Normal Curve Airline Booking Fish in Lake Area of parallelogram Area of triangle Area of trapezoid Triangle with top removed An angle-side correspondence Congruent quadrilaterals Not congruent parallelograms Congruent triangles? Two congruent right triangles First proof of the Pythagorean Theorem Two congruent right triangles Second proof of the Pythagorean Theorem Heron s formula Similar regular hexagons A regular 24-gon The area of a circle A square inscribed in a circle A square circumscribed outside a circle Tetrahedron sliced with plane parallel to base Equal volumes A rectangular parallelepiped to be sliced Sliced rectangular parallelepiped A half rectangular parallelepiped to be sliced Sliced half of a rectangular parallelepiped

13 LIST OF FIGURES xiii 13.25A generalized cone Cones in a sphere Cross-section of a sphere

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15 List of Tables 5.1 Counting Numbers Long division Pair of dice outcomes Twenty-five students Offspring genotype = BB Irreducible polynomials? A random variable distribution Another random variable distribution Craps payoff distribution Powerball Payoff Test Scores Values of A(z) Oscar Handicapping Three Contests Twelve Contests Points-to-Students Lines-to-Contests Addition Table Multiplication Table Another Tournament xv

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17 Chapter 1 Number Sequences In this chapter, we will describe various kinds of sequences of numbers. We will concentrate on the two most important kinds of sequences: arithmetic and geometric. From these, we will learn the two fundamental ways of describing sequences: explicit formulas and recursions. We will also learn how to sum the terms in these sequences. Finally, we will describe some other common sequences, including the Fibonacci numbers. Many other sequences will appear in subsequent chapters. 1.1 Recursions for Arithmetic and Geometric Sequences One of the first things we learn about in mathematics is sequences of numbers. These sequences can be very simple, for instance, the counting numbers: or the even numbers: {1, 2, 3,... } {2, 4, 6, 8,... }. Some are more complex, such as the powers of 10: {1, 10, 100, 1000, 10000,... }, the perfect squares: or the prime numbers: {1, 4, 9, 16, 25,... } {2, 3, 5, 7, 11, 13,... }. Some include negative numbers: { 1, 2, 3, 4, 5, 6, 7, 8,... } 1

18 2 CHAPTER 1. NUMBER SEQUENCES and some include fractions: {1, 1/2, 1/3, 1/4, 1/5,... }. Some can be quite difficult to understand. For instance, the digits of π: {3, 1, 4, 1, 5, 9,... }. At this point, we should issue a warning about our notation. We have given only the first few numbers in the sequence and have left it for you to guess the pattern. In our examples thus far, this pattern has been obvious. Sometimes this is not the case. And even if the pattern seems obvious, we may not have the obvious sequence in mind. For instance, we all assume that the next number in the sequence {1, 2, 3,... } (1.1) is 4. However, suppose this sequence is really or or even {1, 2, 3, 1, 2, 3, 1, 2, 3,... } {1, 2, 3, 3, 2, 1, 1, 2, 3, 3, 2, 1,... } {1, 2, 3, 5, 8, 13, 21,... } For now, we will assume that the sequence 1.1 means the obvious sequence of counting numbers. In this and the following sections, we will concentrate on a special kind of sequence and will only occasionally look at other, more general, sequences. You are already familiar with many sequences of this kind the counting numbers: {1, 2, 3,... }, the even numbers: multiples of 5: multiples of 10: powers of 10: and powers of 2: {2, 4, 6, 8,... }, {5, 10, 15, 20,... }, {10, 20, 30,... }, {1, 10, 100, 1000, 10000,... } {1, 2, 4, 8, 16, 32,... }.

19 1.1. RECURSIONS 3 We are going to consider these and other such sequences in a very general way. The key observation is that each of these sequences is created either by successive addition (counting numbers, even numbers, multiples of 5 or multiples of 10) or by successive multiplication (powers of 10, powers of 2). Successive additions can give us number sequences other than just multiples of a given number. For example, the odd numbers {1, 3, 5,... } are constructed by successively adding 2. Similarly, successive multiplications can give us number sequences other than powers of a given number. For example, {3, 6, 12, 24, 48,... } is constructed by successively multiplying by 2. Number sequences which are formed by successive additions of the same amount are called arithmetic sequences. Those which are formed by successive multiplication by the same amount are called geometric sequences. That is, the difference between successive numbers in an arithmetic sequence is always the same, while the quotient of successive numbers in a geometric sequence is always the same. The even numbers, the odd numbers, and the multiples of 5 are examples of arithmetic sequences, while the powers of 2, the powers of 10 and the sequence {3, 6, 12, 24, 48,... } are examples of geometric sequences. Exercise Is the sequence arithmetic or geometric? Exercise Is the sequence arithmetic or geometric? {2, 10, 50, 250,... } {10, 13, 16, 19,... } Exercise For each of the following sequences, determine if the sequence is arithmetic, geometric, both arithmetic and geometric, or neither arithmetic nor geometric. Also, what is the next number in the sequence? i. {0, 1, 2, 3, 4,... }; ii. {3, 3, 3, 3,... }; iii. {1, 1/2, 1/4, 1/8,... }; iv. {+1, 1, +1, 1,... };

20 4 CHAPTER 1. NUMBER SEQUENCES v. {1, 1/2, 1/3, 1/4,... }; vi. { 1 2, 2, 3 1 2, 5, 6 1 2,... }; vii. {4/5, 2/15, 8/15, 6/5, 28/15,... }; viii. { 3, 3, 3 3, 9, 9 3,... }; ix. {+1, 2, +3, 4, +5,... }. We can display number sequences in a symbolic way. We will let the letter a with subscripts represent a sequence. That is, a 1 represents the first number in the sequence, a 2 the second number, etc. (The use of a is not special. We could have chosen b or c or α.) This gives us a convenient shorthand for number sequences. We may now refer to the entire sequence by writing {a n }. The a n here is a generic member of the sequence. It refers to the nth number (called term) in the sequence. The letter n is called an index. It is also sometimes called a subscript or a parameter. There is nothing special about the use of n as the index. The sequence {a k } is the same as the sequence {a n }. Here is an example. Let a 1 = 3, a 2 = 6, a 3 = 12, a 4 = 24, etc. Then is the sequence Instead of writing {a 1, a 2, a 3,... } {3, 6, 12, 24, 48,... }. {3, 6, 12, 24, 48,... } every time we wish to refer to this sequence, we now simply write {a n }. This notation also gives us a handy way to refer to individual terms in the sequence. In {a n } just described, a 1 = 3, a 5 = 48 and a 11 = Exercise Let {a n } be the sequence What is a 2? a 5? {2, 10, 50, 250,... }. Exercise Let {b n } be the sequence What is b 3? b 6? {3, 8, 13, 18,... }. Exercise Let {c n } be the sequence What is c 3? c 7? { 3, 3, 3 3, 9, 9 3,... }.

21 1.1. RECURSIONS 5 Every arithmetic sequence can be described as follows: the nth term is computed by adding some fixed constant (called the common difference) to the preceding (or (n 1)st) term. For example, if {w n } = {1, 4, 7, 10,... }, then w 7 is w (and w 6 is w 5 + 3, w 5 is w 4 + 3, etc.) and, more generally, w n = w n (1.2) This description is called a recursion. A recursion is a formula for computing a term in a sequence from earlier terms in the sequence. In this example, the only earlier term we need is the preceding term. Exercise Use the recursion in Equation 1.2 to compute w 5, w 6 and w 7. Notice that the choice of n and n 1 as the subscripts in the recursion is somewhat arbitrary. We are simply trying to say, using symbols, that the next term in the sequence is gotten from the previous term by adding three. We might also have written w n+1 = w n + 3 to mean the same thing. Similarly, the nth term of a geometric sequence can be described as some fixed multiple (called the common ratio) of the (n 1)st term. For example, if {t n } = {3, 6, 12, 24, 48,... }, then t 5 = 2t 4 (and t 4 = 2t 3, t 3 = 2t 2, etc.) and, more generally, As above, we might also have written t n = 2t n 1. (1.3) t n+1 = 2t n. Exercise Use the recursion in Equation 1.3 to compute t 6, t 7 and t 8. Our goal is to find a recursion for a general arithmetic sequence and for a general geometric sequence. Exercise For each of the sequences below, determine if the sequence is arithmetic or geometric, find the common difference or ratio, and find a recursion. i. {u n } = {2, 4, 6, 8,... }; ii. {v n } = {1, 3, 5, 7,... }; iii. {s n } = {1, 2, 4, 8, 16,... };

22 6 CHAPTER 1. NUMBER SEQUENCES iv. {α n } = {1/2, 2, 7/2, 5, 13/2,... }; v. {β n } = { 3 2, 2, 2 3 4, 4 3 2, 8,... }; vi. {γ n } = {4/3, 1/4, 11/6, 41/12, 5,... }; vii. {A n } = {π, 3π, 5π, 7π,... }. Exercise Find a recursion for a general arithmetic sequence {a n } with common difference d. Exercise Find a recursion for a general geometric sequence {g n } with common ratio r. and Note that the sequences {u n } = {2, 4, 6, 8,... }, {v n } = {1, 3, 5, 7,... } have the same recursions, although they are different sequences. More information, besides the recursion, is needed to define the sequence unambiguously. For arithmetic and geometric sequences, that information is the value of a single term, called an initial condition. In most cases, that single term is the first term of the sequence. Thus the sequence {w n } is completely defined by the recursion and the initial condition w n = w n w 1 = 1. Similarly, the sequence {t n } is completely defined by t n = 2t n 1 and the initial condition t 1 = 3. Exercise Write recursions and initial conditions for the sequences below i. {2, 7, 12, 17, 22,... }; ii. {2, 6, 18, 54, 162,... }; iii. {3, 30, 300, 3000,... }. Sequences which are both arithmetic and geometric are very special. The next two exercises classify them.

23 1.1. RECURSIONS 7 Exercise Find some sequences which are both arithmetic and geometric. Exercise Describe all sequences which are both arithmetic and geometric. Give an argument justifying your description. Arithmetic and geometric sequences can be used as building blocks for other arithmetic and geometric sequences. The next few exercises show one such kind of construction. Exercise For the sequence write down the sequence and the sequence {v n } = {1, 3, 5, 7,... }, {v 1, v 3, v 5, v 7,... } {v 5, v 8, v 11, v 14,... }. Are these sequences arithmetic? Do the same for the sequence {w n } = {1, 4, 7, 10,... }. Exercise Suppose {a n } is an arithmetic sequence. sequence {a 1, a 3, a 5, a 7,... } is also an arithmetic sequence. Show that the sequence Show that the {a 5, a 8, a 11, a 14,... } is an arithmetic sequence. State and show as general a result of this type as you can. Exercise Repeat Exercise , replacing the word arithmetic with geometric everywhere. Exercise In Exercise we saw that by selecting some of the terms in an arithmetic sequence we get a new arithmetic sequence. Is it possible to select terms from an arithmetic sequence to get a geometric sequence? Try to do this using the even numbers as your arithmetic sequence. Find as many (if any) geometric sequences as you can. Describe all of the geometric sequences that you get. Exercise Can you select terms from a geometric sequence that produce an arithmetic sequence? Try to do this using the powers of 2 as your geometric sequence. Are there any? Why or why not? What can you say in general?

24 8 CHAPTER 1. NUMBER SEQUENCES It is possible to describe many sequences, not just arithmetic and geometric, with recursions. For example, if then Exercise Find ρ 6 and ρ 7. ρ n = 2ρ n 2 ρ n 1, ρ 1 = 1, ρ 2 = 2, ρ 3 = 2ρ 1 ρ 2 = = 0 ρ 4 = 2ρ 2 ρ 3 = = 4 ρ 5 = 2ρ 3 ρ 4 = = 4. Exercise Write the first six terms of the sequences defined by the following recursions and initial conditions. i. a n = a n 1 a n 2 1 with a 1 = 1 and a 2 = 2. ii. b n = b n 1 b n 2 1 with b 1 = 1 and b 2 = 3. iii. c n = c n 1 + n 2 n with c 1 = Explicit Formulas for Arithmetic and Geometric Sequences While a recursion is a useful description of a sequence, it will not be much help in computing, say, the 100th term in a sequence. However, we already have an intuitive idea of how to do this for arithmetic and geometric sequences. Exercise If 2 is the first even number, what is the 12th even number? If 5 is the first multiple of 5, what is the 112th multiple of 5? If 2 is the first power of 2, what is the 13th power of 2? If 3 is the first multiple of 3, what is the nth multiple of 3? Exercise If 1 is the first odd number, what is the 17th odd number? What is the 97th number in the sequence which begins {2, 7, 12, 17, 22,... }? What is the 16th number in the sequence {2, 6, 18, 54, 162,... }? What is the nth number in the sequence {2, 5, 8, 11,... }?

25 1.2. EXPLICIT FORMULAS 9 The last part of these two exercises should have convinced you that there is a formula, involving n, for the nth term in an arithmetic (or geometric) sequence. Such a formula is called an explicit formula. Our goal in this section is to find the explicit formulas for general arithmetic and geometric sequences. For example, if the sequence is then the explicit formula is {w n } = {1, 4, 7, 10,... }, w n = 3n 2. To check this, note that w 1 = = 1, w 2 = = 4, etc. If the sequence is then the explicit formula is {t n } = {3, 6, 12, 24, 48,... }, t n = 3 2 n 1. To check this, note that t 1 = = 3, w 2 = = 6, etc. Exercise Find explicit formulas for each of the following sequences. Arithmetically verify your formulas for n = 1, 2 and 3. i. {u n } = {2, 4, 6, 8,... }; ii. {v n } = {1, 3, 5, 7,... }; iii. {s n } = {1, 2, 4, 8, 16,... }. Exercise Find explicit formulas for each of the following sequences. i. {2, 7, 12, 17, 22,... }; ii. {2, 6, 18, 54, 162,... }; iii. {3, 30, 300, 3000,... }. Exercise Write down an explicit formula for a general arithmetic sequence with common difference d. Exercise Write down an explicit formula for a general geometric sequence with a common ratio r. Exercise If a sequence has an explicit formula which follows your rule for arithmetic/geometric sequences, is it arithmetic/geometric? Why? Exercise For each of the following sequences, find an explicit formula. i. {u n } = {2, 4, 6, 8,... }; ii. {v n } = {1, 3, 5, 7,... };

26 10 CHAPTER 1. NUMBER SEQUENCES iii. {s n } = {1, 2, 4, 8, 16,... }; iv. {α n } = {1/2, 2, 7/2, 5, 13/2,... }; v. {β n } = { 3 2, 2, 2 3 4, 4 3 2, 8,... }; vi. {γ n } = {4/3, 1/4, 11/6, 41/12, 5,... }; vii. {A n } = {π, 3π, 5π, 7π,... }. Recursions and explicit formulas are the two most common ways to define precisely a number sequence. Exercise Explain the difference between a recursion and an explicit formula. Exercise Which of the following formulas is an explicit formula and which is a recursion? a n = n 2 + a 1 + a 10, for n > 10, b k = k + b 1 + b b k 1, for k > 1, c n = n 2, for n 1. The explicit formulas can be used in several ways to solve problems involving arithmetic and geometric sequences. Exercise If the sixth term in an arithmetic sequence is 100 and the first term is 3, what is the 20th term? Exercise If the nth term in an arithmetic sequence is 570, the first term is 3, and that the fifth term is 31, what is n? Exercise If second term in a geometric sequence is 9 and the fifth term is 3, what is the tenth term? Arithmetic and geometric are only two kinds of sequences. We can think of many other number sequences. Some of these also have explicit formulas. For example, the explicit formula yields the sequence t n = n2 1 n {t n } = {0, 3/5, 4/5, 15/17, 12/13,... }.

27 1.2. EXPLICIT FORMULAS 11 Exercise Find an explicit formula for each of these sequences: { 0 1, 1 2, 2 3, 3 4,... }, {0, 3, 8, 15, 24, 35, 48, 63,... }. *Exercise Find an explicit formula for this sequence: {0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4,... } (Hint: use base 2 logarithms). We can now summarize the recursions and explicit formulas for arithmetic and geometric sequences. For an arithmetic sequence {a n } with common difference d, the recursion is a n = a n 1 + d while the explicit formula is a n = a 1 + d(n 1). For a geometric sequence {g n } with common ratio r, the recursion is while the explicit formula is g n = rg n 1 g n = g 1 r n 1. The term a 1 in the sequence {a n } refers to the first term in the sequence. That is, the sequence is {a 1, a 2, a 3,... }. However, sometimes, as a matter of convenience, we want the first term to be indexed by 0, so that the sequence is {b 0, b 1, b 2,... }. For example, we might describe the sequence as {3, 6, 12, 24, 48,... } {a 1, a 2, a 3,... }, so that a 1 = 3, a 2 = 6, etc. But we could also describe it as {b 0, b 1, b 2,... } with b 0 = 3, b 1 = 6, etc. When there could be confusion about what index is used for the first term in the sequence {a n }, we write {a n } n=1,2,....

28 12 CHAPTER 1. NUMBER SEQUENCES Exercise If the sequence is find b 4, b 6 and b n. Exercise If the sequence is find c 4, c 6 and c n. {b k } k=0,1,2,... = {4, 7, 10,... }, {c l } l=0,1,2,... = {4, 6, 9, 27/2,... }, Exercise What happens to the recursion for an arithmetic sequence if the indexing begins at 0 instead of 1? Exercise What happens to the explicit formula for an arithmetic sequence if the indexing begins at 0 instead of 1? 1.3 Summing Arithmetic Sequences Arithmetic and geometric sequences form the building blocks for other interesting sequences. For example, we can create a new sequence by adding the terms in an arithmetic sequence. Suppose {a n } is an arithmetic sequence. What can we say about the sequences {s n } where In the special case that s 1 = a 1, s 2 = a 1 + a 2, s 3 = a 1 + a 2 + a 3, s 4 = a 1 + a 2 + a 3 + a 4,.? {a n } = {1, 2, 3, 4,... }, there is a simple formula for the corresponding s n and a nice proof. In this case, s 1 = 1, s 2 = = 3, s 3 = = 6, s 4 = = 10 and s 5 = = 15. The formula for the sum of the first n counting numbers is s n = n = n(n + 1)/2. (1.4)

29 1.3. SUMMING ARITHMETIC SEQUENCES 13 Exercise Check that Equation (1.4) is correct for the five values s 1, s 2, s 3, s 4, and s 5 above. Figure 1.1 describes s 1, s 2, s 3 and s 4. The black dots in each row represent the counting numbers. The first diagram represents s 1 = 1. The second diagram represents s 2 = The third diagram represents s 3 = , etc. Figure 1.1: Triangular numbers Exercise Use Figure 1.1 to give a proof of Equation (1.4). Because of the diagrams in Figure 1.1, the sequence {s n } is called the sequence of triangular numbers. The sequence of sums can be computed for every arithmetic sequence. Here are two approaches: i. Write the sum forwards and backwards and add corresponding terms. ii. Use the formula for the counting sequence given above. For example, suppose we want to sum the first twenty terms of the arithmetic sequence {9, 13, 17,... }. The twentieth term is 9 + 4(20 1) = 85, so the sum is Now write this sum forwards and backwards and add corresponding terms: = = = So or 2( ) = = 940.

30 14 CHAPTER 1. NUMBER SEQUENCES For the second method, we will manipulate the sum so as to be able to use Equation (1.4): = (9 + 0) + (9 + 4) + (9 + 8) + + ( ) = ( ) = ( ) = = = 940. Notice that Equation (1.4) was used for n = 19, even though the problem was to find the sum of the first twenty terms of the sequence. This is because the counting sequence which emerged in the calculation began at 0, not 1. Exercise Find the sum of the first 100 terms of the arithmetic sequence using both methods. {3, 4, 5, 6,... } Exercise Find the sum of the first 75 terms of the arithmetic sequence using both methods. {3, 6, 9, 12,... } Exercise Find the sum of the first 50 terms of the arithmetic sequence using both methods. {5, 8, 11, 14,... } Exercise Using the first method, derive the following formula: The formula for the sum of the first n terms of the arithmetic sequence {a n } is s n = n 2 (a 1 + a n ). Exercise From the previous exercise and the explicit formula for an arithmetic sequence, derive the following alternate formula: Another formula for the sum of the first n terms of the arithmetic sequence {a n } with common difference d is n(n 1) s n = na 1 + d. 2

31 1.3. SUMMING ARITHMETIC SEQUENCES 15 The formula for the sum of an arithmetic sequence can be combined with the explicit formula in the previous section to solve the following problems. Exercise If you were told that the nth term in an arithmetic sequence is 570, that the first term is 3, and that the fifth term is 31, what would you say is the sum of these n terms? Exercise Suppose the sum of the first 16 terms of an arithmetic sequence is 440, while the sum of the first 8 terms is 124. What is the 16th term? Exercise What is the sum of the last 100 terms in the arithmetic sequence {1, 5,..., 2001}? Exercise What is the sum of the odd numbers between 101 and 999, including 101 and 999? Exercise Sum the even numbers from 48 to 98. *Exercise Suppose the second term in an arithmetic sequence is 2 and the nth term is 32, while the sum of the first n terms is 715/2. Find n. Triangular numbers are depicted in the first picture in Figure 1.2 below. For each of the following triangles we are going to count the number of dots on a side and the number of dots enclosed (that is, on the perimeter or in the interior) by the triangle. For the triangle ABC, these numbers are 2 on a side and 3 enclosed. For the triangle ADE, they are 3 and 6. For the triangle AFG, they are 4 and 10. Finally, for the triangle AHI, they are 5 and 15. Let s include the triangle A, which has 1 and 1 as its numbers. The sequence of the number of circles enclosed then begins {1, 3, 6, 10, 15,... } and is evidently the sequence of triangular numbers (since each interior number includes the previous interior number plus the number of dots on the new side). In a similar manner, the second picture in Figure 1.2 depicts the square numbers. The numbers of dots on each side again are 1, 2, 3, 4 and 5, while the numbers of enclosed dots are the perfect squares 1, 4, 9, 16, and 25. The third picture in Figure 1.2 depicts pentagonal numbers. They are {1, 5, 12, 22, 35,... }. The first is 1, the second is 1+4, the third is 1+4+7, the fourth is , etc. Exercise Use the fact that the pentagonal numbers are the sequence of sums of an arithmetic sequence to find a formula for the nth pentagonal number. *Exercise Write down a sequence of pictures which describes hexagonal numbers. Find a formula for them. Compute the formula for the nth k-polygonal number.

32 16 CHAPTER 1. NUMBER SEQUENCES I G E C A B D F H Figure 1.2: Triangular, square and pentagonal numbers

33 1.4. SUMMING GEOMETRIC SEQUENCES Summing Geometric Sequences With a little bit of algebra, we can also compute the sum sequence for geometric sequences. Notice that (r + 1)(r 1) = r 2 1, (r 2 + r + 1)(r 1) = r 3 1, (r 3 + r 2 + r + 1)(r 1) = r 4 1. Exercise Check the algebra in the above equations by doing the polynomial arithmetic. In general, or, equivalently, (r n 1 + r n r 2 + r + 1)(r 1) = r n 1, (1.5) r n 1 + r n r 2 + r + 1 = rn 1 r 1, r 1. (1.6) Exercise Explain why Equation (1.5) is true. Exercise Explain why Equation (1.6) follows from Equation (1.5). Let s use Equation (1.6) to find the sum of the first 10 terms of the geometric sequence {2, 10, 50, 250,... }. Write = 2( ) by factoring out the first term 2. Next, use Equation (1.6) with r = 5 to get Finally, simplify this last expression: 2( ) = 2(510 1) 5 1 2(5 10 1) 5 1 = Exercise Do a similar calculation to compute the sum of the first 20 terms of this sequence. Exercise Based on the preceding discussion and the previous exercise, find a formula for the sum of the first n terms of this sequence. Exercise Derive a formula for the sum of the first 20 terms of a geometric sequence whose first term is g 1 and whose common ratio is r...

34 18 CHAPTER 1. NUMBER SEQUENCES Exercise Derive a formula for the sum of the first n terms of a geometric sequence whose first term is g 1 and whose common ratio is r. If the absolute value of the common ratio is less than 1, then all the terms of the geometric sequence can be added up. The next set of exercises shows how this works. Exercise Find the sum of the first 10 terms of the geometric sequence {1, 1/2, 1/4,... }. Find the sum of the first 100 terms of the same sequence. Do the same things for the geometric sequence {1, 1/3, 1/9,... }. Exercise Find the sum of the first n terms of the geometric sequence {1, 1/2, 1/4,... }. Find the sum of the first n terms of the geometric sequence {1, 1/3, 1/9,... }. Exercise What happens to (1/2) n as n grows large? What happens to the sum of the first n terms of the geometric sequence {1, 1/2, 1/4,... } as n grows large? Exercise What happens to the sum of the first n terms of the geometric sequence {1, 1/3, 1/9,... } as n grows large? From Exercise we have the following. The formula for the sum of the first n terms of the geometric sequence {g n } is s n = g 1 rn 1 r 1. (1.7) Exercise Use Equation 1.7 to prove s n = g 1 1 rn 1 r. Exercise In the equation in Exercise , what happens to r n as n grows large, when r < 1? Show that the sequence of sums, {s 1, s 2, s 3,... }, for a geometric sequence {g n } gets close to a certain value as n gets large, if r < 1. The previous exercise shows that if {g n } is a geometric sequence with common ration r and if r < 1, then g 1 + g 2 + g gets close to a certain value which we will call s. In fact, a consequence of this exercise is the following: The formula for all the terms in a geometric sequence is s = g r.

35 1.5. EXAMPLES 19 Exercise Find the infinite sums 1 + 1/2 + 1/4 + 1/8 + and 1 + 1/3 + 1/9 + 1/27 +. Exercise Find the infinite sums /3 + 8/9 + 16/27 + and 1 1/2 + 1/4 1/8 + 1/16 1/32 +. Exercise Find the infinite sum Examples of Arithmetic and Geometric Sequences Arithmetic and geometric sequences have many applications. Here are several examples. In each example, you will have to determine whether an arithmetic or a geometric sequence is appropriate. Sometimes you will have to sum the sequence. Exercise A ladder with 15 rungs is tapered so that the top rung is 12 inches wide while the bottom rung is 18 inches wide. Find the total length of all the rungs. Exercise Puff pastry is made as follows. The dough is rolled into a rectangle. Then two-thirds of the rectangle is buttered. The unbuttered third is folded over, followed by the other buttered third (like a letter), to get a stack of two butter layers sandwiched between three dough layers. See Figure 1.3. Now this new rectangle of dough is rolled out and the process is repeated. If this buttering, folding and rolling is done six times, how many layers of dough are there in the final pastry? Exercise If a clock chimes the hour on the hour (e.g., it chimes five times at 5 o clock), and it also chimes once on the half-hour, how many times does it chime from 12:15 p.m. to 12:15 p.m. the next day? An important application of geometric sequences is compound interest. Suppose we invest $2 at 5% per year. In the first year, our two dollars earns 10 cents interest. If that dime is invested along with the two dollars for the second year, the original two dollars earns another dime and the dime interest from

36 20 CHAPTER 1. NUMBER SEQUENCES fold 1 fold 2 butter butter Figure 1.3: Puff pastry the first year earns This interest on the interest is called compound interest. So after the second year, we have our original two dollars, two dimes of interest on those two dollars, and $.005 interest on interest. This is the same as since 2 (1 +.05) (1 +.05) = 2 ( ) = = For the third year, we now have invested, which earns 5%. So we have $ = $ after three years. If our interest is compounded every six months, then our $2 earns 2.5% in the first six months, and the $2 plus this interest earns another 2.5% in the next six months. Thus, after one year, we have $ = $ After two years, we will have $ $ And after three years, we will have $ $ Exercise Suppose in 1787 George Washington invested $25 at 4% interest, compounded yearly. What would his descendants have today? What if the compounding was done quarterly instead of yearly? Daily? Every second? At 6 p.m., the minute hand of a standard clock points straight up and the hour hand points straight down. Thirty minutes later, the minute hand catches up to where the hour hand was at 6 p.m. However, the hour hand has moved halfway between the 6 and the 7. Two and one-half minutes later, the minute hand reaches that position, but the hour hand has moved again. Exercise What time is it when the minute hand catches the hour hand? That is, at what time between 6 p.m. and 7 p.m. are the minute hand and the hour hand pointing in exactly the same direction? Solve this problem using an appropriate sequence. Can you find other ways to solve it?