Introduction. Appendix D Mathematical Induction D1


 Gavin Hines
 2 years ago
 Views:
Transcription
1 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 find a linear or quadratic model. Introduction In this section, you will study a form of mathematical proof called mathematical induction. It is important that you clearly see the logical need for it, so take a look at the problem below. S S 3 S S S Judging from the pattern formed by these first five sums, it appears that the sum of the first n odd integers is S n n n. Although this particular formula is valid, it is important for you to see that recognizing a pattern and then simply jumping to the conclusion that the pattern must be true for all values of n is not a logically valid method of proof. There are many examples in which a pattern appears to be developing for small values of n and then at some point the pattern fails. One of the most famous cases of this was the conjecture by the French mathematician Pierre de Fermat (0 5), who speculated that all numbers of the form F n n, are prime. For n 0,,, 3, and, the conjecture is true. F 0 3, F 5, F 7, F 3 57, F 5,537 The size of the next Fermat number F 5,9,97,97 is so great that it was difficult for Fermat to determine whether it was prime or not. However, another wellknown mathematician, Leonhard Euler ( ), later found the factorization F 5,9,97,97,700,7 n 0,,,... which proved that F 5 is not prime and therefore Fermat s conjecture was false. Just because a rule, pattern, or formula seems to work for several values of n, you cannot simply decide that it is valid for all values of n without going through a legitimate proof. Mathematical induction is one method of proof.
2 D Appendix D Mathematical Induction The Principle of Mathematical Induction Let P n be a statement involving the positive integer n. If. P is true, and. the truth of P k implies the truth of P k for every positive integer k, then P n must be true for all positive integers n. It is important to recognize that both parts of the Principle of Mathematical Induction are necessary. To apply the Principle of Mathematical Induction, you need to be able to determine the statement for a given statement P k. Example Using to Find Find for each statement. a. P k : S k k k b. P k : S k k 3 k 3 c. P k : 3 k k SOLUTION a. Replace k by k. b. c. P k P k : S k k k k k P k : 3 k k 3 k k 3 P k P k P k Simplify. P k : S k k ) ] 3 k k 3 k Checkpoint Find for P k : S k kk. P k FIGURE D. A wellknown illustration used to explain why the Principle of Mathematical Induction works is the unending line of dominoes shown in Figure D.. If the line actually contains infinitely many dominoes, then it is clear that you could not knock the entire line down by knocking down only one domino at a time. However, suppose it were true that each domino would knock down the next one as it fell. Then you could knock them all down simply by pushing the first one and starting a chain reaction. Mathematical induction works in the same way. If the truth of P k implies the truth of P k and if P is true, then the chain reaction proceeds as follows: P implies P, P implies P 3, implies P, and so on. P 3
3 Using Mathematical Induction Appendix D Mathematical Induction D3 Example Using Mathematical Induction Use mathematical induction to prove the formula STUDY TIP When using mathematical induction to prove a summation formula (such as the one in Example ), it is helpful to think of S k as S k S k a k, where a k is the k th term of the original sum. S n n n for all integers n. SOLUTION Mathematical induction consists of two distinct parts. First, you must show that the formula is true when n.. When n, the formula is valid, because S. The second part of mathematical induction has two steps. The first step is to assume that the formula is valid for some integer k. The second step is to use this assumption to prove that the formula is valid for the next integer, k.. Assuming that the formula S k k k is true, you must show that the formula S k k is true. S k k k k k S k k Group terms to form S k. k k Replace S k by k. k Factor. Combining the results of parts () and (), you can conclude by mathematical induction that the formula is valid for all integers n. Checkpoint Use mathematical induction to prove the formula S n n 3 for all integers n. nn 7 It occasionally happens that a statement involving natural numbers is not true for the first k positive integers but is true for all values of n k. In these instances, you use a slight variation of the Principle of Mathematical Induction in which you verify P k rather than P. This variation is called the Extended Principle of Mathematical Induction. To see the validity of this variation, note from Figure D. that all but the first k dominoes can be knocked down by knocking over the kth domino. This suggests that you can prove a statement P n to be true for n k by showing that P k is true and that P k implies P k. In Exercises of this section, you are asked to apply this extension of mathematical induction.
4 D Appendix D Mathematical Induction Example 3 Using Mathematical Induction Use mathematical induction to prove the formula S n 3... n for all integers n. SOLUTION. When n, the formula is valid, because. Assuming that nn n S you must show that S k To do this, write the following. S k S k a k kk k k 3. S k 3... k k k k kk k k k kk k k k 7k k k k 3 kk k 3... k k By assumption Add expressions. Factor out k. Simplify. Completely factor. Combining the results of parts () and (), you can conclude by mathematical induction that the formula is valid for all integers n. (k k k 3 Checkpoint 3 Use mathematical induction to prove the formula S n n for all integers n. 3nn When proving a formula by mathematical induction, the only statement that you need to verify is P. As a check, however, it is a good idea to try verifying some of the other statements. For instance, in Example 3, try verifying and S 3. S
5 Appendix D Mathematical Induction D5 Sums of Powers of Integers The formula in Example 3 is one of a collection of useful summation formulas. This and other formulas dealing with the sums of various powers of the first n positive integers are shown. Sums of Powers of Integers n 3... n nn nn n n 3 n n 3... n nn n 3n 3n n 5 n n n n Each of these formulas for sums can be proven by mathematical induction. Example Finding a Sum of Powers of Integers Find 7 n n SOLUTION Using the formula for the sum of the cubes of the first n positive integers, you obtain the following. 7 n n 9 78 Check this sum by adding the numbers, as shown below Checkpoint 7 7 Find 0 n n
6 D Appendix D Mathematical Induction Example 5 Proving an Inequality by Mathematical Induction Prove that n < n for all positive integers n. SOLUTION. For n and n, the formula is true, because < and <.. Assuming that k < k you need to show that k < k. For n k, you have k k > k k. By assumption Because k k k > k for all k >, it follows that k > k > k or k < k. Combining the results of parts () and (), you can conclude by mathematical induction that n < n for all positive integers n. Checkpoint 5 Prove that n! > n for all integers n > 3. Pattern Recognition Although choosing a formula on the basis of a few observations does not guarantee the validity of the formula, pattern recognition is important. Once you have a pattern that you think works, you can try using mathematical induction to prove your formula. Finding a Formula for the nth Term of a Sequence To find a formula for the nth term of a sequence, consider these guidelines.. Calculate the first several terms of the sequence. It is often a good idea to write the terms in both simplified and factored forms.. Try to find a recognizable pattern for the terms and write a formula for the nth term of the sequence. This is your hypothesis or conjecture. You might try computing one or two more terms in the sequence to test your hypothesis. 3. Use mathematical induction to prove your hypothesis.
7 Appendix D Mathematical Induction D7 Example Finding a Formula for a Finite Sum Find a formula for the finite sum SOLUTION Begin by writing out the first few sums. S S 3 3 From this sequence, it appears that the formula for the kth sum is k k. To prove the validity of this hypothesis, use mathematical induction, as shown below. Note that you have already verified the formula for n, so you can begin by assuming that the formula is valid for n k and trying to show that it is valid for n k. S k k k k k k k So, the hypothesis is valid. Checkpoint k k k k kk k k k k k nn S S S k By assumption Add fractions. Distributive Property Factor numerator. Simplify. kk kk k k Find a formula for the finite sum n
8 D8 Appendix D Mathematical Induction Finite Differences The first differences of a sequence are found by subtracting consecutive terms. The second differences are found by subtracting consecutive first differences. The first and second differences of the sequence 3, 5, 8,, 7, 3,... are as shown. n: a n : First differences: Second differences: For this sequence, the second differences are all the same nonzero number. When this happens, the sequence has a perfect quadratic model. When the first differences are all the same, the sequence has a linear model. That is, it is arithmetic Example 7 Finding an Appropriate Model Decide whether the sequence,,, 8, 78,,... can be represented perfectly by a linear or a quadratic model. If so, find the model. SOLUTION Find the first and second differences of the sequence. n: 3 5 a n : 8 78 First differences: Second differences: Because the second differences are all the same, the sequence can be represented perfectly by a quadratic model of the form a n an bn c. By substituting,, and 3 for n, you can obtain a system of three linear equations in three variables. a a b c a a b c a 3 a3 b3 c Substitute for n. Substitute for n. Substitute 3 for n. You now have a system of three equations in a, b, and c. a b c Equation a b c Equation 9a 3b c Equation 3 Using the techniques discussed in Chapter 5, you can find that the solution of this system is a, b, and c 8. So, the quadratic model is a n n n 8. Checkpoint 7 Decide whether the sequence 3, 8, 7, 30, 7, 8,... can be represented perfectly by a linear or a quadratic model. If so, find the model.
9 Appendix D Mathematical Induction D9 Example 8 Finding an Appropriate Model Decide whether the sequence, 9,, 3, 30, 37,... can be represented perfectly by a linear or a quadratic model. If so, find the model. SOLUTION Find the first differences of the sequence. n: 3 5 a n : First differences: Because, the first differences are all the same, the sequence can be represented perfectly by a linear model of the form a n an b. By substituting and for n, you can obtain a system of two linear equations in two variables. a a b a a b 9 Substitute for n. Substitute for n. You now have a system of two equations in a and b. a b Equation a b 9 Equation Using the techniques discussed in Chapter 5, you can find that the solution of this system is a 7 and b 5. So, the linear model is a n 7n 5. Checkpoint 8 Decide whether the sequence, 0,, 8,,,... can be represented perfectly by a linear or a quadratic model. If so, find the model. SUMMARIZE. State the principle of mathematical induction (page D). For examples of using the principle of mathematical induction to prove a formula, see Examples and 3.. Make a list of the formulas for the sums of powers of integers (page D5). For an example that uses a formula to find a sum of powers of integers, see Example. 3. Describe the guidelines for finding a formula for the nth term of a sequence (page D). For an example of finding a formula for the nth term of a sequence, see Example.. Describe how to use the finite differences of a sequence to determine whether the sequence can be represented perfectly by a linear or a quadratic model (page D8). For examples of using finite differences to find models, see Examples 7 and 8.
10 D0 Appendix D Mathematical Induction SKILLS WARM UP D The following warmup exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. In Exercises, find the sum k 3 5 j j. k k3 j In Exercises 5 0, simplify the expression. k 3 3k k 3 k 3 8. k k k k 50 k i i Exercises D Using to find In Exercises, find for the given P k. See Example P k. P k kk k k 3 3. k P. P k k k k k Using Mathematical Induction In Exercises 5 8, use mathematical induction to prove the formula for every positive integer n. See Examples and n nn n nn P k n ii i P k n n n 3 n nn n n 3 n n n n n i 5 n n n n i n i nn n 3n 3n i 30 nn n 3 nn P k n 3 n 5n n n 3n n n 8. Finding a Sum of Powers of Integers In Exercises 9 8, find the sum using the formulas for the sums of powers of integers. See Example. 9. n n n. 7. i 8i 3 8. Checking Solutions In Exercises 9 3, determine whether the inequality is true for n,, 3, and. 9. n < n n n 3. n n 3 3. n 3 < n n 33. n n 3. < n! n n n Proving an Inequality by Mathematical Induction In Exercises 35 38, prove the inequality for the indicated integer values of n. See Example n! > n, n n i 0 n n n 5 n n n i i i 3... n > n, x y n y n, n n 50 n n 0 n 3 n 8 n 5 n 0 n 3 n n 5 j 3 j j 3 n > n, 38. < x n and 0 < x < y n 7 n
11 Appendix D Mathematical Induction D Finding a Formula for a Finite Sum In Exercises 39, find a formula for the sum of the first n terms of the sequence. See Example. 39., 5, 9, 3, ,, 9,,...., 9 0, 8 00, ,.... 3, 9, 7, 8 8,... 3.,,, 0,..., nn,.... 3, 3, 5, 5,..., n n,... Proving Properties In Exercises 5 5, use mathematical induction to prove the given property for all positive integers n. 5. ab n a n b n. 7. If x then x x x. 0,. x. 0,..., x n 0, 3 x n x x x 3... x n. 8. If x > 0, then lnx x x. x.. > 0,..., x n > 0, 3 x n ln x ln x ln x 3... ln xn. 9. Generalized Distributive Law: xy y... y n xy xy... xy n 50. a bi n and a bi n are complex conjugates for all n. 5. A factor of n 3 3n n is A factor of n 3 n is Writing In your own words, explain what is meant by a proof by mathematical induction. 5. Think About It What conclusion can be drawn from the given information about the sequence of statements P n? (a) P 3 is true and P k implies P k. (b) P, P, P 3,..., P 50 are all true. (c) P, P, and P 3 are all true, but the truth of P k does not imply the truth of P k. (d) P is true and implies P k. P k Finding an Appropriate Model In Exercises 55 0, decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, find the model. See Examples 7 and , 3,, 9, 37, 5,... 5., 9,, 3, 30, 37, , 5, 30, 5, 78,, ,,, 30, 8, 70, ,,, 3,, 33,... 0., 8, 3,, 7, 0,... a b n an b n a 0 0 Using Finite Differences In Exercises 70, write the first five terms of the sequence, where a f. Then calculate the first and second differences of the sequence. Does the sequence have either a linear model or a quadratic model? If so, find the model.. f 0. f a n a n 3 a n n a n 3. f 3. f 3 a n a n n a n a n 5. a 0 0. a 0 a n a n n a n a n 7. f 8. f 0 a n a n a n a n n 9. a a n a n n a n a n Finding a Quadratic Model In Exercises 7 7, find a quadratic model for the sequence with the indicated terms. 7. a 0 3, a 3, a 5 7. a 0 7, a, a a 0, a 8, a a 0 3, a 5, a Think About It When the second differences of a sequence are all zero, can the sequence be represented perfectly by a linear model? Explain. 7. HOW DO YOU SEE IT? The table shows the distance d s (in feet) required to stop a truck from the moment the brakes are applied at a speed of s milesperhour on dry, level pavement. The first and second differences of the sequence of terms d s are shown below the table. (Source: Virginia Transportation Research Council) Speed, s Distance, d s First differences: Second differences: (a) Will a linear or a quadratic model fit the data best? Explain. (b) When you drive a truck, do you assume an equal risk for each 0 mileperhour increase in speed? Explain your reasoning.
Appendix F: Mathematical Induction
Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another
More informationSECTION 102 Mathematical Induction
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 0.. 6. Approximate e 0.5 using the first five terms
More information8.7 Mathematical Induction
8.7. MATHEMATICAL INDUCTION 8135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture
More informationSection 62 Mathematical Induction
6 Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that
More informationWorksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation
Worksheet on induction MA113 Calculus I Fall 2006 First, let us explain the use of for summation. The notation f(k) means to evaluate the function f(k) at k = 1, 2,..., n and add up the results. In other
More informationLecture 3. Mathematical Induction
Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion
More informationMATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:
MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,
More informationMath 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
More informationMathematical Induction
Mathematical Induction MAT30 Discrete Mathematics Fall 016 MAT30 (Discrete Math) Mathematical Induction Fall 016 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)
More informationp 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)
.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other
More informationMathematical Induction
Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural
More informationMATHEMATICAL 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
More informationDirect Proofs. CS 19: Discrete Mathematics. Direct Proof: Example. Indirect Proof: Example. Proofs by Contradiction and by Mathematical Induction
Direct Proofs CS 19: Discrete Mathematics Amit Chakrabarti Proofs by Contradiction and by Mathematical Induction At this point, we have seen a few examples of mathematical proofs. These have the following
More informationBasic 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
More informationChapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1
Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes
More informationMathematical Induction. Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition)
Mathematical Induction Rosen Chapter 4.1 (6 th edition) Rosen Ch. 5.1 (7 th edition) Mathmatical Induction Mathmatical induction can be used to prove statements that assert that P(n) is true for all positive
More information3. Recurrence Recursive Definitions. To construct a recursively defined function:
3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a
More informationMathematical Induction
Chapter 4 Mathematical Induction 4.1 The Principle of Mathematical Induction Preview Activity 1 (Exploring Statements of the Form.8n N/.P.n//) One of the most fundamental sets in mathematics is the set
More informationPRINCIPLE OF MATHEMATICAL INDUCTION
Chapter 4 PRINCIPLE OF MATHEMATICAL INDUCTION Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction Newton was indebted to it for his theorem
More information2.4 Mathematical Induction
2.4 Mathematical Induction What Is (Weak) Induction? The Principle of Mathematical Induction works like this: What Is (Weak) Induction? The Principle of Mathematical Induction works like this: We want
More informationMATHEMATICAL INDUCTION
MATHEMATICAL INDUCTION MATH 5A SECTION HANDOUT BY GERARDO CON DIAZ Imagine a bunch of dominoes on a table. They are set up in a straight line, and you are about to push the first piece to set off the chain
More informationClimbing an Infinite Ladder
Section 5.1 Climbing an Infinite Ladder Suppose we have an infinite ladder and the following capabilities: 1. We can reach the first rung of the ladder. 2. If we can reach a particular rung of the ladder,
More informationMathematical 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,
More information1.2. 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
More informationHOMEWORK 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
More informationINTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS
INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28
More informationMath 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
More informationDEFINITIONS: 1. FACTORIAL: n! = n (n 1) (n 2) ! = 1. n! = n (n 1)! = n (n 1) (n 2)! (n + 1)! = (n + 1) n! = (n + 1) n (n 1)!
MATH 2000 NOTES ON INDUCTION DEFINITIONS: 1. FACTORIAL: n! = n (n 1) (n 2)... 3 2 1 0! = 1 n! = n (n 1)! = n (n 1) (n 2)! (n + 1)! = (n + 1) n! = (n + 1) n (n 1)! 2. SUMMATION NOTATION: f(i) = f(1) + f(2)
More informationCongruences. Robert Friedman
Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition
More informationCourse Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction.
Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. February 21, 2006 1 Proof by Induction Definition 1.1. A subset S of the natural numbers is said to be inductive if n S we have
More informationAnswer 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.
More informationPrinciple of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))
Outline We will cover (over the next few weeks) Mathematical Induction (or Weak Induction) Strong (Mathematical) Induction Constructive Induction Structural Induction Principle of (Weak) Mathematical Induction
More informationk, 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
More informationProof of Infinite Number of Fibonacci Primes. Stephen Marshall. 22 May Abstract
Proof of Infinite Number of Fibonacci Primes Stephen Marshall 22 May 2014 Abstract This paper presents a complete and exhaustive proof of that an infinite number of Fibonacci Primes exist. The approach
More information3. 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)
More informationInduction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition
Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing
More informationMATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS
1 CHAPTER 6. MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 Introduction Recurrence
More informationCONTINUED 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.
More informationCHAPTER 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
More information8.3. GEOMETRIC SEQUENCES AND SERIES
8.3. GEOMETRIC SEQUENCES AND SERIES What You Should Learn Recognize, write, and find the nth terms of geometric sequences. Find the sum of a finite geometric sequence. Find the sum of an infinite geometric
More informationWe now explore a third method of proof: proof by contradiction.
CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement
More informationPractice Problems for First Test
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
More informationIt is time to prove some theorems. There are various strategies for doing
CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it
More informationSolutions 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
More informationRunning head: A GEOMETRIC INTRODUCTION 1
Running head: A GEOMETRIC INTRODUCTION A Geometric Introduction to Mathematical Induction Problems using the Sums of Consecutive Natural Numbers, the Sums of Squares, and the Sums of Cubes of Natural Numbers
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationIB Maths SL Sequence and Series Practice Problems Mr. W Name
IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =
More informationStanford University Educational Program for Gifted Youth (EPGY) Number Theory. Dana Paquin, Ph.D.
Stanford University Educational Program for Gifted Youth (EPGY) Dana Paquin, Ph.D. paquin@math.stanford.edu Summer 2010 Note: These lecture notes are adapted from the following sources: 1. Ivan Niven,
More information3.3 MATHEMATICAL INDUCTION
Section 3.3 Mathematical Induction 3.3.1 3.3 MATHEMATICAL INDUCTION From modus ponens: p p q q basis assertion conditional assertion conclusion we can easily derive double modus ponens : p 0 p 0 p 1 p
More informationSummer High School 2009 Aaron Bertram. 1. Prime Numbers. Let s start with some notation:
Summer High School 2009 Aaron Bertram 1. Prime Numbers. Let s start with some notation: N = {1, 2, 3, 4, 5,...} is the (infinite) set of natural numbers. Z = {..., 3, 2, 1, 0, 1, 2, 3,...} is the set of
More informationThis is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0).
This is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More informationSTUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS
STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an
More informationMathematical Induction. Lecture 1011
Mathematical Induction Lecture 1011 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
More informationHomework 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
More informationDiscrete Mathematics, Chapter 5: Induction and Recursion
Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Wellfounded
More informationSEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Alessandro Artale UniBZ  http://www.inf.unibz.it/ artale/ SECTION 5.2 Mathematical Induction I Copyright Cengage Learning. All rights reserved.
More informationSan Jose Math Circle October 17, 2009 ARITHMETIC AND GEOMETRIC PROGRESSIONS
San Jose Math Circle October 17, 2009 ARITHMETIC AND GEOMETRIC PROGRESSIONS DEFINITION. An arithmetic progression is a (finite or infinite) sequence of numbers with the property that the difference between
More information9.2 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
More informationIndiana 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
More informationElementary 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
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Note 3
CS 70 Discrete Mathematics and Probability Theory Fall 06 Seshia and Walrand Note 3 Mathematical Induction Introduction. In this note, we introduce the proof technique of mathematical induction. Induction
More information4. PRINCIPLE OF MATHEMATICAL INDUCTION
4 PRINCIPLE OF MATHEMATICAL INDUCTION Ex Prove the following by principle of mathematical induction 1 1 + 2 + 3 + + n 2 1 2 + 2 2 + 3 2 + + n 2 3 1 3 + 2 3 + 3 3 + + n 3 + 4 (1) + (1 + 3) + (1 + 3 + 5)
More informationLESSON 1 PRIME NUMBERS AND FACTORISATION
LESSON 1 PRIME NUMBERS AND FACTORISATION 1.1 FACTORS: The natural numbers are the numbers 1,, 3, 4,. The integers are the naturals numbers together with 0 and the negative integers. That is the integers
More informationDiscrete 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
More information1.3 Induction and Other Proof Techniques
4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.
More informationAPPLICATIONS OF THE ORDER FUNCTION
APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and
More informationCSE373: Data Structures and Algorithms Lecture 2: Proof by Induction. Linda Shapiro Winter 2015
CSE373: Data Structures and Algorithms Lecture 2: Proof by Induction Linda Shapiro Winter 2015 Background on Induction Type of mathematical proof Typically used to establish a given statement for all natural
More informationAdvanced Algebra 2. I. Equations and Inequalities
Advanced Algebra 2 I. Equations and Inequalities A. Real Numbers and Number Operations 6.A.5, 6.B.5, 7.C.5 1) Graph numbers on a number line 2) Order real numbers 3) Identify properties of real numbers
More informationSample 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
More information22 Proof Proof questions. This chapter will show you how to:
22 Ch22 537544.qxd 23/9/05 12:21 Page 537 4 4 3 9 22 537 2 2 a 2 b This chapter will show you how to: tell the difference between 'verify' and 'proof' prove results using simple, stepbystep chains of
More informationEvery 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
More informationHandout #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
More informationCOMP232  Mathematics for Computer Science
COMP3  Mathematics for Computer Science Tutorial 10 Ali Moallemi moa ali@encs.concordia.ca Iraj Hedayati h iraj@encs.concordia.ca Concordia University, Winter 016 Ali Moallemi, Iraj Hedayati COMP3  Mathematics
More informationExam #5 Sequences and Series
College of the Redwoods Mathematics Department Math 30 College Algebra Exam #5 Sequences and Series David Arnold Don Hickethier Copyright c 000 DonHickethier@Eureka.redwoods.cc.ca.us Last Revision Date:
More informationMathematical Induction with MS
Mathematical Induction 200814 with MS 1a. [4 marks] Using the definition of a derivative as, show that the derivative of. 1b. [9 marks] Prove by induction that the derivative of is. 2a. [3 marks] Consider
More informationPrimes 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
More informationStudents in their first advanced mathematics classes are often surprised
CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are
More information4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY
PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY 1 Oh the things you should learn How to recognize and write arithmetic sequences
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationMathematical Induction
Mathematical S 0 S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 Like dominoes! Mathematical S 0 S 1 S 2 S 3 S4 S 5 S 6 S 7 S 8 S 9 S 10 Like dominoes! S 4 Mathematical S 0 S 1 S 2 S 3 S5 S 6 S 7 S 8 S 9 S 10
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More information1.3 Algebraic Expressions
1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,
More informationThe 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,
More informationAlgebra 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,
More information3. QUADRATIC CONGRUENCES
3. QUADRATIC CONGRUENCES 3.1. Quadratics Over a Finite Field We re all familiar with the quadratic equation in the context of real or complex numbers. The formula for the solutions to ax + bx + c = 0 (where
More information(3n 2) = (3n 2) = 1(3 1) 2 1 = (3k 2) =
Question 1 Prove using mathematical induction that for all n 1, Solution 1 + + 7 + + (n ) = n(n 1) For any integer n 1, let P n be the statement that 1 + + 7 + + (n ) = n(n 1) Base Case The statement P
More informationAPPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic
More information2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.
2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationIn a triangle with a right angle, there are 2 legs and the hypotenuse of a triangle.
PROBLEM STATEMENT In a triangle with a right angle, there are legs and the hypotenuse of a triangle. The hypotenuse of a triangle is the side of a right triangle that is opposite the 90 angle. The legs
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationSequences and Mathematical Induction. CSE 215, Foundations of Computer Science Stony Brook University
Sequences and Mathematical Induction CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 Sequences A sequence is a function whose domain is all the integers
More informationRevised 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.
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1Semester 2 Grade Level: 1012 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More information8 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.
More informationPRINCIPLE OF MATHEMATICAL INDUCTION
Chapter 4 PRINCIPLE OF MATHEMATICAL INDUCTION 4.1 Overview Mathematical induction is one of the techniques which can be used to prove variety of mathematical statements which are formulated in terms of
More informationExponents and Radicals
Exponents and Radicals (a + b) 10 Exponents are a very important part of algebra. An exponent is just a convenient way of writing repeated multiplications of the same number. Radicals involve the use of
More informationCalculus for Middle School Teachers. Problems and Notes for MTHT 466
Calculus for Middle School Teachers Problems and Notes for MTHT 466 Bonnie Saunders Fall 2010 1 I Infinity Week 1 How big is Infinity? Problem of the Week: The Chess Board Problem There once was a humble
More informationUndergraduate 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
More informationArithmetic and Geometric Sequences
Arithmetic and Geometric Sequences Felix Lazebnik This collection of problems is for those who wish to learn about arithmetic and geometric sequences, or to those who wish to improve their understanding
More information