# SECTION 10-2 Mathematical Induction

Size: px
Start display at page:

Transcription

1 73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e Approximate e 0.5 using the first five terms of the series. Compare this approximation with your calculator evaluation of e Show that: n k 64. Show that: n k ca k c n a k k (a k b k ) n k a k n b k k SECTION 0- Mathematical Induction Introduction Mathematical Induction Additional Examples of Mathematical Induction Three Famous Problems Introduction In common usage, the word induction means the generalization from particular cases or facts. The ability to formulate general hypotheses from a limited number of facts is a distinguishing characteristic of a creative mathematician. The creative process does not stop here, however. These hypotheses must then be proved or disproved. In mathematics, a special method of proof called mathematical induction ranks among the most important basic tools in a mathematician s toolbox. In this section mathematical induction will be used to prove a variety of mathematical statements, some new and some that up to now we have just assumed to be true. We illustrate the process of formulating hypotheses by an example. Suppose we are interested in the sum of the first n consecutive odd integers, where n is a positive integer. We begin by writing the sums for the first few values of n to see if we can observe a pattern: n 3 4 n n n n 5 Is there any pattern to the sums, 4, 9, 6, and 5? You no doubt observed that each is a perfect square and, in fact, each is the square of the number of terms in the sum. Thus, the following conjecture seems reasonable: Conjecture P n : For each positive integer n, (n ) n That is, the sum of the first n odd integers is n for each positive integer n. So far ordinary induction has been used to generalize the pattern observed in the first few cases listed above. But at this point conjecture P n is simply that a conjecture. How do we prove that P n is a true statement? Continuing to list specific cases will never provide a general proof not in your lifetime or all your descendants lifetimes! Mathematical induction is the tool we will use to establish the validity of conjecture P n.

2 0- Mathematical Induction 733 Before discussing this method of proof, let s consider another conjecture: Conjecture Q n : For each positive integer n, the number n n 4 is a prime number. TABLE n n n 4 Prime? 4 Yes 43 Yes 3 47 Yes 4 53 Yes 5 6 Yes It is important to recognize that a conjecture can be proved false if it fails for only one case. A single case or example for which a conjecture fails is called a counterexample. We check the conjecture for a few particular cases in Table. From the table, it certainly appears that conjecture Q n has a good chance of being true. You may want to check a few more cases. If you persist, you will find that conjecture Q n is true for n up to 4. What happens at n 4? which is not prime. Thus, since n 4 provides a counterexample, conjecture Q n is false. Here we see the danger of generalizing without proof from a few special cases. This example was discovered by Euler ( ). EXPLORE-DISCUSS Prove that the following statement is false by finding a counterexample: If n, then at least one-third of the positive integers less than or equal to n are prime. Mathematical Induction We begin by stating the principle of mathematical induction, which forms the basis for all our work in this section. Theorem Principle of Mathematical Induction Let P n be a statement associated with each positive integer n, and suppose the following conditions are satisfied:. P is true.. For any positive integer k, if is true, then is also true. Then the statement P n is true for all positive integers n. Theorem must be read very carefully. At first glance, it seems to say that if we assume a statement is true, then it is true. But that is not the case at all. If the two conditions in Theorem are satisfied, then we can reason as follows: P is true. Condition P is true, because P is true. Condition P 3 is true, because P is true. Condition P 4 is true, because P 3 is true. Condition..

3 734 0 Sequences and Series Condition : The first domino can be pushed over. (a) Since this chain of implications never ends, we will eventually reach P n for any positive integer n. To help visualize this process, picture a row of dominoes that goes on forever (see Fig. ) and interpret the conditions in Theorem as follows: Condition says that the first domino can be pushed over. Condition says that if the kth domino falls, then so does the (k )st domino. Together, these two conditions imply that all the dominoes must fall. Now, to illustrate the process of proof by mathematical induction, we return to the conjecture P n discussed earlier, which we restate below: P n : (n ) n n any positive integer Condition : If the kth domino falls, then so does the (k )st. (b) We already know that P is a true statement. In fact, we demonstrated that P through P 5 are all true by direct calculation. Thus, condition in Theorem is satisfied. To show that condition is satisfied, we assume that is a true statement: : (k ) k Now we must show that this assumption implies that is also a true statement: : (k ) (k ) (k ) Conclusion: All the dominoes will fall. (c) Since we have assumed that is true, we can perform operations on this equation. Note that the left side of is the left side of plus (k ). So we start by adding (k ) to both sides of : FIGURE Interpreting mathematical induction (k ) k (k ) (k ) k (k ) Factoring the right side of this equation, we have Add k to both sides (k ) (k ) (k ) But this last equation is. Thus, we have started with, the statement we assumed true, and performed valid operations to produce, the statement we want to be true. In other words, we have shown that if is true, then is also true. Since both conditions in Theorem are satisfied, P n is true for all positive integers n. Additional Examples of Mathematical Induction Now we will consider some additional examples of proof by induction. The first is another summation formula. Mathematical induction is the primary tool for proving that formulas of this type are true. EXAMPLE Proving a Summation Formula Prove that for all positive integers n n n n

4 0- Mathematical Induction 735 Proof Part State the conjecture: Show that P is true. P n : n n n P : Thus, P is true. Part Show that if is true, then is true. It is a good practice to always write out both and at the beginning of any induction proof to see what is assumed and what must be proved: : : k k k k k k k We assume is true. We must show that follows from. We start with the true statement, add / k to both sides, and simplify the right side: k k k k k k k k k k k k k k k Thus, k k k k and we have shown that if is true, then is true. Conclusion Both conditions in Theorem are satisfied. Thus, P n is true for all positive integers n.

5 736 0 Sequences and Series Matched Problem Prove that for all positive integers n 3... n n(n ) The next example provides a proof of a law of exponents that previously we had to assume was true. First we redefine a n for n a positive integer, using a recursion formula: DEFINITION Recursive Definition of a n For n a positive integer a a a n a n a n EXAMPLE Proving a Law of Exponents Prove that (xy) n x n y n for all positive integers n. Proof State the conjecture: P n : (xy) n x n y n Part Show that P is true. (xy) xy x y Definition Definition Thus, P is true. Part Show that if is true, then is true. : (xy) k x k y k : (xy) k x k y k Assume is true. Show that follows from. Here we start with the left side of and use to find the right side of : (xy) k (xy) k (xy) x k y k xy (x k x)(y k y) x k y k Definition Use : (xy) k x k y k Property of real numbers Definition

6 0- Mathematical Induction 737 Thus, (xy) k x k y k, and we have shown that if is true, then is true. Conclusion Both conditions in Theorem are satisfied. Thus, P n is true for all positive integers n. Matched Problem Prove that (x/y) n x n /y n for all positive integers n. Our last example deals with factors of integers. Before we start, recall that an integer p is divisible by an integer q if p qr for some integer r. EXAMPLE 3 Proving a Divisibility Property Prove that 4 n is divisible by 5 for all positive integers n. Proof Use the definition of divisibility to state the conjecture as follows: P n : 4 n 5r for some integer r Part Show that P is true. P : Thus, P is true. Part Show that if is true, then is true. : 4 k 5r for some integer r Assume is true. : 4 (k ) 5s for some integer s Show that must follow. As before, we start with the true statement : 4 k 5r 4 (4 k ) 4 (5r) 4 k 6 80r 4 (k ) 80r 5 5(6r 3) Multiply both sides by 4. Simplify. Add 5 to both sides. Factor out 5. Thus, 4 (k ) 5s where s 6r 3 is an integer, and we have shown that if is true, then is true.

7 738 0 Sequences and Series Conclusion Both conditions in Theorem are satisfied. Thus, P n is true for all positive integers n. Matched Problem 3 Prove that 8 n is divisible by 7 for all positive integers n. In some cases, a conjecture may be true only for n m, where m is a positive integer, rather than for all n 0. For example, see Problems 49 and 50 in Exercise 0-. The principle of mathematical induction can be extended to cover cases like this as follows: Theorem Extended Principle of Mathematical Induction Let m be a positive integer, let P n be a statement associated with each integer n m, and suppose the following conditions are satisfied:. P m is true.. For any integer k m, if is true, then is also true. Then the statement P n is true for all integers n m. Three Famous Problems The problem of determining whether a certain statement about the positive integers is true may be extremely difficult. Proofs may require remarkable insight and ingenuity and the development of techniques far more advanced than mathematical induction. Consider, for example, the famous problems of proving the following statements:. Lagrange s Four Square Theorem, 77: Each positive integer can be expressed as the sum of four or fewer squares of positive integers.. Fermat s Last Theorem, 637: For n, x n y n z n does not have solutions in the natural numbers. 3. Goldbach s Conjecture, 74: Every positive even integer greater than is the sum of two prime numbers. The first statement was considered by the early Greeks and finally proved in 77 by Lagrange. Fermat s last theorem, defying the best mathematical minds for over 350 years, finally succumbed to a 00-page proof by Prof. Andrew Wiles of Princeton University in 993. To this date no one has been able to prove or disprove Goldbach s conjecture. EXPLORE-DISCUSS (A) Explain the difference between a theorem and a conjecture. (B) Why is Fermat s last theorem a misnomer? Suggest more accurate names for the result.

8 0- Mathematical Induction 739 Answers to Matched Problems. Sketch of proof. State the conjecture: P n : 3... n ( ) Part.. P is true. Part. Show that if is true, then is true. n(n ) Conclusion: P n is true.. Sketch of proof. State the conjecture: P n : x y x y x 3... k(k ) k 3... k (k ) x y n x n y n Part.. P is true. y Part. Show that if is true, then is true. k(k ) (k ) (k )(k ) x y k x y k x y x k y y k x xk x y k y xk y k Conclusion: P n is true. 3. Sketch of proof. State the conjecture: P n : 8 n 7r for some integer r Part P is true. Part. Show that if is true, then is true. 8 k 7r 8(8 k ) 8(7r) 8 k 56r 7 7(8r ) 7s Conclusion: P n is true. EXERCISE 0- A In Problems 4, find the first positive integer n that causes the statement to fail.. 3 n 4 n 5 n. n 3n n is divisible by n 4. n 5n 6 Verify each statement P n in Problems 5 0 for n,, and P n : (4n ) n 6. P n : n n(n ) 7. P n : a 5 a n a 5 n 8. P n : (a 5 ) n a 5n 9. P n : 9 n is divisible by 4 0. P n : 4 n is divisible by 3 Write and for P n as indicated in Problems 6.. P n in Problem 5. P n in Problem 6 3. P n in Problem 7 4. P n in Problem 8 5. P n in Problem 9 6. P n in Problem 0 In Problems 7, use mathematical induction to prove that each P n holds for all positive integers n. 7. P n in Problem 5 8. P n in Problem 6 9. P n in Problem 7 0. P n in Problem 8. P n in Problem 9. P n in Problem 0

9 740 0 Sequences and Series B In Problems 3 6, prove the statement is false by finding a counterexample. 3. If n, then any polynomial of degree n has at least one real zero. 4. Any positive integer n 7 can be written as the sum of three or fewer squares of positive integers. 5. If n is a positive integer, then there is at least one prime number p such that n p n If a, b, c, d are positive integers such that a b c d, then a c or a d. In Problems 7 4, use mathematical induction to prove each proposition for all positive integers n, unless restricted otherwise n n n n (n ) 3 (4n 3 n) (n ) (n ) ; n n n(n )(n ) n(n ) n(n )(n ) 3 a n a a n 3 ; n ; n 5 a a 3 n a n a m a n a m n ; m, n N [Hint: Choose m as an arbitrary element of N, and then use induction on n.] 36. (a n ) m a mn ; m, n N 37. x n is divisible by x ; x [Hint: Divisible means that x n (x )Q(x) for some polynomial Q(x).] 38. x n y n is divisible by x y; x y 39. x n is divisible by x ; x 40. x n is divisible by x ; x n 3 ( 3... n) [Hint: See Matched Problem following Example.] 4. C In Problems 43 46, suggest a formula for each expression, and prove your hypothesis using mathematical induction, n N n n(n ) 45. The number of lines determined by n points in a plane, no three of which are collinear 46. The number of diagonals in a polygon with n sides In Problems 47 50, prove the statement is true for all integers n as specified. 47. a a n ; n N a 0 a n ; n N 49. n n; n n n ; n 5 5. Prove or disprove the generalization of the following two facts: Prove or disprove: n n is a prime number for all natural numbers n. If {a n } and {b n } are two sequences, we write {a n } {b n } if and only if a n b n, n N. In Problems 53 56, use mathematical induction to show that {a n } {b n }. 53. a, a n a n ; b n n 54. a, a n a n ; b n n 55. a, a n a n ; b n n 56. a, a n 3a n ; b n 3 n n(n )(n ) n(n 3) 4(n )(n ) SECTION 0-3 Arithmetic and Geometric Sequences Arithmetic and Geometric Sequences nth-term Formulas Sum Formulas for Finite Arithmetic Series Sum Formulas for Finite Geometric Series Sum Formula for Infinite Geometric Series

### Introduction. Appendix D Mathematical Induction D1

Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to

### 1.2. Successive Differences

1. An Application of Inductive Reasoning: Number Patterns In the previous section we introduced inductive reasoning, and we showed how it can be applied in predicting what comes next in a list of numbers

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Basic Proof Techniques

Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

### Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

### Mathematical Induction. Mary Barnes Sue Gordon

Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

### Theorem3.1.1 Thedivisionalgorithm;theorem2.2.1insection2.2 If m, n Z and n is a positive

Chapter 3 Number Theory 159 3.1 Prime Numbers Prime numbers serve as the basic building blocs in the multiplicative structure of the integers. As you may recall, an integer n greater than one is prime

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### Math 115 Spring 2011 Written Homework 5 Solutions

. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### Primes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov

Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

### Mathematical Induction. Lecture 10-11

Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

### 2.1. Inductive Reasoning EXAMPLE A

CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers

### Sample Induction Proofs

Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### WRITING PROOFS. Christopher Heil Georgia Institute of Technology

WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

### The thing that started it 8.6 THE BINOMIAL THEOREM

476 Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers (b) Based on your results for (a), guess the minimum number of moves required if you start with an arbitrary number of n disks.

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### 4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

### Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

### MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.

MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

### Section 6-3 Arithmetic and Geometric Sequences

466 6 SEQUENCES, SERIES, AND PROBABILITY Section 6- Arithmetic and Geometric Sequences Arithmetic and Geometric Sequences nth-term Formulas Sum Formulas for Finite Arithmetic Series Sum Formulas for Finite

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### APPENDIX 1 PROOFS IN MATHEMATICS. A1.1 Introduction 286 MATHEMATICS

286 MATHEMATICS APPENDIX 1 PROOFS IN MATHEMATICS A1.1 Introduction Suppose your family owns a plot of land and there is no fencing around it. Your neighbour decides one day to fence off his land. After

### SOLVING EQUATIONS WITH RADICALS AND EXPONENTS 9.5. section ( 3 5 3 2 )( 3 25 3 10 3 4 ). The Odd-Root Property

498 (9 3) Chapter 9 Radicals and Rational Exponents Replace the question mark by an expression that makes the equation correct. Equations involving variables are to be identities. 75. 6 76. 3?? 1 77. 1

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### 4.2 Euclid s Classification of Pythagorean Triples

178 4. Number Theory: Fermat s Last Theorem Exercise 4.7: A primitive Pythagorean triple is one in which any two of the three numbers are relatively prime. Show that every multiple of a Pythagorean triple

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system

CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### SCORE SETS IN ORIENTED GRAPHS

Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Ummmm! Definitely interested. She took the pen and pad out of my hand and constructed a third one for herself:

Sum of Cubes Jo was supposed to be studying for her grade 12 physics test, but her soul was wandering. Show me something fun, she said. Well I wasn t sure just what she had in mind, but it happened that

### Session 6 Number Theory

Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

### Five fundamental operations. mathematics: addition, subtraction, multiplication, division, and modular forms

The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms UC Berkeley Trinity University March 31, 2008 This talk is about counting, and it s about

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### 5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1

MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### Cubes and Cube Roots

CUBES AND CUBE ROOTS 109 Cubes and Cube Roots CHAPTER 7 7.1 Introduction This is a story about one of India s great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### Negative Integer Exponents

7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions

### This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

### Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

### C H A P T E R Regular Expressions regular expression

7 CHAPTER Regular Expressions Most programmers and other power-users of computer systems have used tools that match text patterns. You may have used a Web search engine with a pattern like travel cancun

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Number Theory Hungarian Style. Cameron Byerley s interpretation of Csaba Szabó s lectures

Number Theory Hungarian Style Cameron Byerley s interpretation of Csaba Szabó s lectures August 20, 2005 2 0.1 introduction Number theory is a beautiful subject and even cooler when you learn about it

### Congruent Number Problem

University of Waterloo October 28th, 2015 Number Theory Number theory, can be described as the mathematics of discovering and explaining patterns in numbers. There is nothing in the world which pleases

### Grade 7/8 Math Circles Sequences and Series

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Sequences and Series November 30, 2012 What are sequences? A sequence is an ordered

### Our Primitive Roots. Chris Lyons

Our Primitive Roots Chris Lyons Abstract When n is not divisible by 2 or 5, the decimal expansion of the number /n is an infinite repetition of some finite sequence of r digits. For instance, when n =

### Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014

Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### SECTION A-3 Polynomials: Factoring

A-3 Polynomials: Factoring A-23 thick, write an algebraic epression in terms of that represents the volume of the plastic used to construct the container. Simplify the epression. [Recall: The volume 4

### PROOFS BY DESCENT KEITH CONRAD

PROOFS BY DESCENT KEITH CONRAD As ordinary methods, such as are found in the books, are inadequate to proving such difficult propositions, I discovered at last a most singular method... that I called the

### Factoring and Applications

Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

### MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

### CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

### 3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.

SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid

### Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

### Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

### Mechanics 1: Vectors

Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized

### Vocabulary. Term Page Definition Clarifying Example. biconditional statement. conclusion. conditional statement. conjecture.

CHAPTER Vocabulary The table contains important vocabulary terms from Chapter. As you work through the chapter, fill in the page number, definition, and a clarifying example. biconditional statement conclusion

### Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### Warm up. Connect these nine dots with only four straight lines without lifting your pencil from the paper.

Warm up Connect these nine dots with only four straight lines without lifting your pencil from the paper. Sometimes we need to think outside the box! Warm up Solution Warm up Insert the Numbers 1 8 into

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### Factoring Algorithms

Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors