Pierre de Fermat ( )

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Section 04 Mathematical Induction 987 8 Find the sum of the first ten terms of the sequence: 9 Find the sum of the first 50 terms of the sequence: 0 Find the sum of the first ten terms of the

1 Section 04 Mathematical Induction Find the sum of the first ten terms of the sequence: 9 Find the sum of the first 50 terms of the sequence: 0 Find the sum of the first ten terms of the sequence: Find the sum of the first 00 terms of the sequence: In Exercises 5, find each indicated sum 4 a i + 4i - a i - i = 4 a a i 5 b a i = 5, 0, 0, 40, Á -, 0,, 4, Á -0, 40, -80, 0, Á 4, -, -8, -4, Á 50 i = q i = i - a - 5 b Express 045 as a fraction in lowest terms 7 Express the sum using summation notation Use i for the index of summation Á A skydiver falls feet during the first second of a dive, 48 feet during the second second, 80 feet during the third second, feet during the fourth second, and so on Find the distance that the skydiver falls during the 5th second and the total distance the skydiver falls in 5 seconds 9 If the average value of a house increases 0% per year, how much will a house costing $0,000 be worth in 0 years? Round to the nearest dollar Section Objectives Understand the principle of mathematical induction Prove statements using mathematical induction 04 Mathematical Induction Pierre de Fermat (0 5) was a lawyer who enjoyed studying mathematics In a margin of one of his books, he claimed that no positive integers satisfy x n + y n = z n if n is an integer greater than or equal to If n =, we can find positive integers satisfying x n + y n = z n, or x + y = z : + 4 = 5 However, Fermat claimed that no positive integers satisfy x + y = z, x 4 + y 4 = z 4, x 5 + y 5 = z 5, After ten years of work, Princeton University s Andrew Wiles proved Fermat s Last Theorem and so on Fermat claimed to have a proof of his conjecture, but added, The margin of my book is too narrow to write it down Some believe that he never had a proof and intended to frustrate his colleagues In 994, 40-year-old Princeton math professor Andrew Wiles proved Fermat s Last Theorem using a principle called mathematical induction In this section, you will learn how to use this powerful method to prove statements about the positive integers Understand the principle of mathematical induction The Principle of Mathematical Induction How do we prove statements using mathematical induction? Let s consider an example We will prove a statement that appears to give a correct formula for the sum of the first n positive integers: S n : Á nn + + n =

2 988 Chapter 0 Sequences, Induction, and Probability We can verify S n : Á + n = integers If n =, the statement is nn + for, say, the first four positive Take the first term on the left =? (+) Substitute for n on the right # = This true statement shows that is true If n =, the statement is S Add the first two terms on the left +=? (+) Substitute for n on the right # = This true statement shows that is true S If n =, the statement is S Add the first three terms on the left ++=? (+) Substitute for n on the right # 4 = This true statement shows that is true S Finally, if n = 4, the statement is S 4 Add the first four terms on the left +++4=? 4(4+) Substitute 4 for n on the right 0 4 # 5 0 = 0 This true statement shows that is true S 4 This approach does not prove that the given statement is true for every positive integer n The fact that the formula produces true statements for n =,,, and 4 does not guarantee that it is valid for all positive integers n Thus, we need to be able to verify the truth of S n without verifying the statement for each and every one of the positive integers A legitimate proof of the given statement S n involves a technique called mathematical induction S n

3 Section 04 Mathematical Induction 989 The Principle of Mathematical Induction Let S n be a statement involving the positive integer n If is true, and the truth of the statement implies the truth of the statement +, for every positive integer k, then the statement S n is true for all positive integers n The principle of mathematical induction can be illustrated using an unending line of dominoes, as shown in Figure 07 If the first domino is pushed over, it knocks down the next, which knocks down the next, and so on, in a chain reaction To topple all the dominoes in the infinite sequence, two conditions must be satisfied: Figure 07 Falling dominoes illustrate the principle of mathematical induction The first domino must be knocked down If the domino in position k is knocked down, then the domino in position k + must be knocked down If the second condition is not satisfied, it does not follow that all the dominoes will topple For example, suppose the dominoes are spaced far enough apart so that a falling domino does not push over the next domino in the line The domino analogy provides the two steps that are required in a proof by mathematical induction The Steps in a Proof by Mathematical Induction Let S n be a statement involving the positive integer n To prove that S n is true for all positive integers n requires two steps Step Show that is true Step Show that if is assumed to be true, then + is also true, for every positive integer k Notice that to prove S n, we work only with the statements,, and + Our first example provides practice in writing these statements EXAMPLE Writing,, and For the given statement S n, write the three statements,, and + a S n : Á nn + + n = b S n : Á nn + n + + n = Solution a We begin with S n : Á + n = nn + Write by taking the first term on the left and replacing n with on the right + : =

4 990 Chapter 0 Sequences, Induction, and Probability S n : Á nn + + n = The statement for part (a) (repeated) Write by taking the sum of the first k terms on the left and replacing n with k on the right : Á + k = Write + by taking the sum of the first k + terms on the left and replacing n with k + on the right + : Á + k + = + : Á + k + = kk + k + k k + k + Simplify on the right b We begin with Write by taking the first term on the left and replacing n with on the right S n : Á + n = Using S n : Á nn + n + + n =, we write by taking the sum of the first k terms on the left and replacing n with k on the right : Á + k = Write + by taking the sum of the first k + terms on the left and replacing n with k + on the right + : Á + k + = : = + # + nn + n + kk + k + k + k + + 4k : Á + k + = k + k + k + Simplify on the right Check Point For the given statement and + a b Á + n = nn Á + n = n n + S n, write the three statements,, Always simplify + before trying to use mathematical induction to prove that S n is true For example, consider S n : Á + n - = Begin by writing + The sum of the first k + terms as follows: ± : [(k+)-] 4 (k+)[(k+)-][(k+)+] = Replace n with k + on the right side of S n nn - n +

5 Section 04 Mathematical Induction 99 Prove statements using mathematical induction Now simplify both sides of the equation + : Á + k + - = + : Á + k + = Proving Statements about Positive Integers Using Mathematical Induction k + k + - k + + k + k + k + Now that we know how to find,, and +, let s see how we can use these statements to carry out the two steps in a proof by mathematical induction In Examples and, we will use the statements,, and + to prove each of the statements S n that we worked with in Example EXAMPLE Proving a Formula by Mathematical Induction Use mathematical induction to prove that for all positive integers n Solution Step Show that is true Statement is Simplifying on the right, we obtain = This true statement shows that is true Step Show that if is true, then is true Using and + from Example (a), show that the truth of, implies the truth of +, Á + n = = Á + k = Á + k + = nn + kk +, k + k + We will work with Because we assume that is true, we add the next consecutive integer after k namely, k + to both sides Á kk + This is, which we + k = assume is true k(k+) +++ +k+(k+)= +(k+) We do not have to write this k because k is understood to be the integer that precedes k Á kk + k + + k + = Á + k + = Á + k + = k + k + k + k + Add k + to both sides of the equation Write the right side with a common denominator of Factor out the common factor k + on the right This final result is the statement +

6 99 Chapter 0 Sequences, Induction, and Probability We have shown that if we assume that is true and we add k + to both sides of, then + is also true By the principle of mathematical induction, the statement S n, namely, Á nn + + n =, is true for every positive integer n Check Point Use mathematical induction to prove that Á + n = nn + for all positive integers n EXAMPLE Proving a Formula by Mathematical Induction Use mathematical induction to prove that for all positive integers n Solution Step Show that is true Statement is Simplifying, we obtain Further simplification on the right gives the statement = This true statement shows that is true Step Show that if is true, then is true Using and from Example (b), show that the truth of implies the truth of Á + k = Á + k + k + = Á + n = = # # = + # + : Á + k = + : Á + k + = nn + n + kk + k + + k + k + k + We will work with Because we assume that is true, we add the square of the next consecutive integer after k, namely, k +, to both sides of the equation kk + k + kk + k Á kk + k + k + + k + = + k + = This is, assumed to be true We must work with this and show is true + + k + Add k + to both sides It is not necessary to write on the left Express the right side with the least common denominator, kk + + k + 4 Factor out the common factor k + k = k + k + 7k + Multiply and combine like terms

7 Section 04 Mathematical Induction 99 = k + k + k + Factor k + 7k + = k + k + k + This final statement is + We have shown that if we assume that S is true, and we add k + k to both sides of, then + is also true By the principle of mathematical induction, the statement S n, namely, Á nn + n + + n =, is true for every positive integer n Check Point Use mathematical induction to prove that Á + n = n n + for all positive integers n Example 4 illustrates how mathematical induction can be used to prove statements about positive integers that do not involve sums 4 EXAMPLE 4 Using the Principle of Mathematical Induction Prove that is a factor of n + 5n for all positive integers n Solution Step Show that is true Statement reads Simplifying the arithmetic, the statement reads is a factor of This statement is true: that is, = # This shows that is true Step Show that if is true, then is true Let s write and + : We can rewrite statement as follows: Statement + : is a factor of k + 5k + : is a factor of k + + 5k + now reads + is a factor of + 5 # by simplifying the algebraic expression in the statement (k+) +5(k+)=k +k++5k+5=k +7k+ Use the formula (A + B) = A + AB + B is a factor of k + 7k + We need to use statement S that is, is a factor of k k + 5k to prove statement + We do this as follows: k +7k+=(k +5k)+(k+)=(k +5k)+(k+) We assume that is a factor of k + 5k because we assume is true Factoring the last two terms shows that is a factor of k +

8 994 Chapter 0 Sequences, Induction, and Probability k +7k+=(k +5k)+(k+)=(k +5k)+(k+) We assume that is a factor of k + 5k because we assume is true Factoring the last two terms shows that is a factor of k + Exercise Set 04 Practice Exercises In Exercises 5 0, a statement about the positive integers is given Write statements and +, simplifying statement completely 5 S n : Á + 4n = nn + S n : Á nn n + = 7 S n : Á + 4n - = nn + 8 S n : Á n5n - + 5n - = 9 S is a factor of n n : - n + 0 S is a factor of n n : - n In Exercises 4, use mathematical induction to prove that each statement is true for every positive integer n Á + 4n = nn Á nn n + = Á + n - = n Á nn + + n = Á + 4n - = nn Á n5n - + 5n - = Á + n - = n - S n We ve repeated the equation from the bottom of the previous page The voice balloons show that is a factor of k + 5k and of k + Thus, if is true, is a factor of the sum k + 5k + k +, or of k + 7k + This is precisely statement + We have shown that if we assume that is true, then + is also true By the principle of mathematical induction, the statement S n, namely is a factor of n + 5n, is true for every positive integer n Check Point 4 Prove that is a factor of n + n for all positive integers n Á + n - = n - In Exercises 4, a statement about the positive integers is given S n Write statements, S, and S, and show that each of these statements is true Á + n = n + - S n : Á + n - = n Á + n = - n S n : Á nn n + = S is a factor of n n : - n 4 S is a factor of n n : - n + 4 Practice Plus In Exercises 5 4, use mathematical induction to prove that each statement is true for every positive integer n 5 is a factor of n - n is a factor of n + n 7 is a factor of nn + n + 8 is a factor of nn + n nn + n + # + # + # 4 + Á + nn + = nn + n + 7 # + # 4 + # 5 + Á + nn + = # + # + # + Á + 4 # + # # 5 + Á + n a 5 # i = n - i = n a 7 # 8 i = 88 n - i = n + 7 n If 0 x, then 0 x n ab n = a n b n 4 a a n b b = an b n nn + = n n + n + n + = n n + 4

9 Section 04 Mathematical Induction 995 Writing in Mathematics 5 Explain how to use mathematical induction to prove that a statement is true for every positive integer n Consider the statement S n given by Although, S, Á, S 40 are true, S 4 is false Verify that is false Then describe how this is illustrated by the dominoes in the figure What does this tell you about a pattern, or formula, that seems to work for several values of n? S 5 S S 7 S 8 n - n + 4 is prime Critical Thinking Exercises S 9 S 40 S 4 S 4 Make Sense? In Exercises 7 40, determine whether each statement makes sense or does not make sense, and explain your reasoning 7 I use mathematical induction to prove that statements are true for all real numbers n 8 I begin proofs by mathematical induction by writing and +, both of which I assume to be true 9 When a line of falling dominoes is used to illustrate the principle of mathematical induction, it is not necessary for all the dominoes to topple 40 This triangular arrangement of circles illustrates that S 4 Some statements are false for the first few positive integers, but true for some positive integer m on In these instances, you can prove S n for n Ú m by showing that S m is true and that implies + when k 7 m Use this extended principle of mathematical induction to prove that each statement in Exercises 4 4 is true 4 Prove that n 7 n + for n Ú Show that the formula is true for n = and then use step of mathematical induction 4 Prove that n 7 n for n Ú 5 Show that the formula is true for n = 5 and then use step of mathematical induction In Exercises 4 44, find through S 5 and then use the pattern to make a conjecture about S n Prove the conjectured formula for S n by mathematical induction 4 44 S n : Á + S n : a - ba - ba - 4 b Á a - Group Exercise nn + =? n + b =? 45 Fermat s most notorious theorem, described in the section opener on page 987, baffled the greatest minds for more than three centuries In 994, after ten years of work, Princeton University s Andrew Wiles proved Fermat s Last Theorem People magazine put him on its list of the 5 most intriguing people of the year, the Gap asked him to model jeans, and Barbara Walters chased him for an interview Who s Barbara Walters? asked the bookish Wiles, who had somehow gone through life without a television Using the 99 PBS documentary Solving Fermat: Andrew Wiles or information about Andrew Wiles on the Internet, research and present a group seminar on what Wiles did to prove Fermat s Last Theorem, problems along the way, and the role of mathematical induction in the proof is true for n = Á + n = nn + Preview Exercises Exercises 4 48 will help you prepare for the material covered in the next section Each exercise involves observing a pattern in the expanded form of the binomial expression a + b n a + b = a + b a + b = a + ab + b a + b = a + a b + ab + b a + b 4 = a 4 + 4a b + a b + 4ab + b 4 a + b 5 = a 5 + 5a 4 b + 0a b + 0a b + 5ab 4 + b 5 4 Describe the pattern for the exponents on a 47 Describe the pattern for the exponents on b 48 Describe the pattern for the sum of the exponents on the variables in each term

Appendix G: Mathematical Induction

Appendix G: 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