Introduction. Appendix D Mathematical Induction D1


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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.
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