Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Mathematical Induction Learn to prove statements by mathematical induction. SECTION

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THE PRINCIPLE OF MATHEMATICAL INDUCTION Let P n be a statement that involves the natural number n with the following properties: 1.P 1 is true (the statement is true for the natural number 1), and 2.If P k is true statement, the P k+1 is a true statement. Then the statement P n is true for every natural number n.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DETERMINING THE STATEMENT P k +1 FROM THE STATEMENT P k Suppose the given statement is P k : k ≥ 1; That is, P k+1 asserts the same property for k + 1 that P k asserts for k. P k+1 : k + 1 ≥ 1. then we have

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using Mathematical Induction Use mathematical induction to prove that, for all natural numbers n, Solution First verify that the statement is true for n = 1. The second condition requires two steps. Step 1 Assume the formula is true for k. 

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using Mathematical Induction Solution continued Begin by using P k, the statement assumed to be true and add 2(k + 1) to both sides Step 2On the basis of the assumption that P k is true, show that P k+1 is true.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Using Mathematical Induction Solution continued This last statement says that P k+1 is true if P k is assumed to be true. Therefore, by the principle of mathematical induction, the statement is true for every natural number n.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Using Mathematical Induction Use mathematical induction to prove that Solution First verify that the statement is true for n = 1. Step 1 Assume the formula is true for k. for every natural number n. Step 2 Use P k to prove that the following is true.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Using Mathematical Induction Solution continued By the product rule of exponents, 2 k+1 = 2 k 2 1, so multiply both sides by 2. Thus, 2 k+1 > k + 1 is true. By mathematical induction, the statement 2 n > n is true for every natural number n.