Mathematical Induction I Lecture 21 Section 4.2 Fri, Feb 23, 2007

The Principle of Mathematical Induction Let P(n) be a predicate defined for integers n. Let a be an integer. If the following two statements are true P(a) For all integers k  a, if P(k), then P(k + 1) then the statement For all integers n  a, P(n) is true.

The Idea The first part shows that the statement is true for the integer a. The second part shows that Since it is true for a, it is true for a + 1. Since it is true for a + 1, it is true for a + 2. Since it is true for a + 2, it is true for a + 3. And so on. Therefore, it is true for all integers  a.

Like Dominos

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Proof by Mathematical Induction Basic Step Choose a starting point a (typically 0 or 1). Prove P(a). Inductive Step Suppose P(k) for some arbitrary integer k  a. Prove P(k + 1), using the assumption P(k). Conclude P(n) for all n  a.

Example: Mathematical Induction Theorem: For any integer n  4, we can obtain n¢ using only 2¢ and 5¢ coins. Let P(n) be the predicate “we can obtain n¢ using only 2¢ and 5¢ coins.” Proof: Basic Step: (Start at a = 4.) P(4) is true since 4¢ = 2¢ + 2¢.

Proof continued Inductive Step Suppose that P(k) is true for some k  4. We must show that P(k + 1) is true. Consider two cases: Case 1: k¢ uses a 5¢ coin. Case 2: k¢ does not use a 5¢ coin.

Proof concluded Case 1: k¢ uses a 5¢ coin. Then remove it and replace it with three 2¢ coins, thereby obtaining (k + 1)¢. Case 2: k¢ does not use a 5¢ coin. Then it must use at least two 2¢ coins. Replace two 2¢ coins with one 5¢ coin, thereby obtaining (k + 1)¢.

Proof concluded Therefore, P(n) is true for all n  4. Therefore, P(k + 1) is true. Therefore, P(n) is true for all n  4.

Example: Mathematical Induction Theorem: For all n  1, Proof: Basic Step When n = 1, we have

Proof and Therefore, the statement is true when n = 1.

Proof Inductive Step Suppose that the statement is true when n = k, for some k  1. That is, suppose that

Proof Then

Proof Therefore, the statement is true for all n  1. Therefore, the statement is true when n = k + 1. Therefore, the statement is true for all n  1.

Example: Mathematical Induction Theorem: For all n  1, Proof: Basic Step Show… Inductive Step Suppose…

Sums of Powers of Integers We can also prove by induction that

Mathematical Induction Mathematical induction requires that we “know” the answer in advance. The method verifies the answer. How would we come up with the guess that in the first place?

Finding the Formula We might conjecture that the answer is a cubic polynomial in n. Why? That is, for some real numbers a, b, c, and d. Then figure out what a, b, c, and d are. How?

Finding the Formula Substitute the values 0, 1, 2, and 3 into the equation to get a system of four equations. d = 0. a + b + c + d = 12 = 1. 8a + 4b + 2c + d = 12 + 22 = 5. 27a + 9b + 3c + d = 12 + 22 + 32 = 14.

Finding the Formula Solve the system of equations and get b = 1/2, c = 1/6, d = 0. Then verify using mathematical induction.

Let’s Play “Find the Flaw” Theorem: For every positive integer n, in any set of n horses, all the horses are the same color. Proof: Basic Step. When n = 1, there is only one horse, so trivially they are (it is) all the same color.

Find the Flaw Inductive Step Suppose that any set of k horses are all the same color. Consider a set of k + 1 horses. Remove one of the horses from the set. The remaining set of k horses are all the same color.

Find the Flaw Replace that horse and remove a different horse. Again, the remaining set of k horses are all the same color. Therefore, the two horses that were removed are the same color as the other horses in the set. Thus, the k + 1 horses are all the same color.

Find the Flaw Thus, in any set of n horses, the horses are all the same color.

The Paradox of the Pop Quiz A professor wants to give a pop quiz to his class on a day when they are not expecting it. Theorem: For all n  0, the professor cannot give the pop quiz n days before the last day. Corollary: The professor cannot give a pop quiz.