1. Chapter 10 Sequences, Induction, and Probability Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 10.4 Mathematical Induction

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Understand the principle of mathematical induction. Prove statements using mathematical induction. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 The Principle of Mathematical Induction Let S n be a statement involving the positive integer n. If 1. S 1 is true, and 2. the truth of the statement S k implies the truth of the statement S k+1, for every positive integer k, then the statement S n is true for all positive integers n.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 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 1 Show that S 1 is true. Step 2 Show that if S k is assumed to be true, then S k+1 is also true, for every positive integer k.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Writing S 1, S k, and S k+1 For the given statement S n, write the three statements S 1, S k, and S k+1. We simplify on the right

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Writing S 1, S k, and S k+1 For the given statement S n, write the three statements S 1, S k, and S k+1. We simplify on the right

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Proving a Formula by Mathematical Induction Use mathematical induction to prove that is true. Step 1 Show that S 1 is true. This true statement shows that S 1 is true.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Proving a Formula by Mathematical Induction (continued) Use mathematical induction to prove that is true. Step 2 Show that if S k is true, then S k+1 is true. We assume that S k is true and add 2(k+1) to both sides, then simplify.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Proving a Formula by Mathematical Induction (continued) Use mathematical induction to prove that is true. Step 2 (cont) Show that if S k is true, then S k+1 is true. We have shown that if we assume that S k is true and add 2(k+1) to both sides, then S k+1 is also true. By the principle of mathematical induction is true for every positive integer n.