896 CHAPTER 12 Sequences; Induction; the Binomial Theorem Now Work PROBLEM 1 CAUTION The conclusion that a statement involving natural numbers is true for all natural numbers is made only after both Conditions I and II of the Principle of Mathematical Induction have been satisfied. Problem 28 demonstrates a statement for which only Condition I holds, and the statement is not true for all natural numbers. Problem 29 demonstrates a statement for which only Condition II holds, and the statement is not true for any natural number. j Using Mathematical Induction Show that − 3 1 n is divisible by 2 for all natural numbers n. Solution EXAMPLE 4 First, show that the statement is true when = n 1. Because − = − = 3 1 3 1 2 1 is divisible by 2, the statement is true when = n 1. Condition I is satisfied. Next, assume that the statement holds for some natural number k, and determine whether the statement holds for the next natural number, + k 1. Assume that − 3 1 k is divisible by 2 for some k. Now show that − + 3 1 k 1 is divisible by 2. ( ) ( ) ( ) − = − + − = − + − = ⋅ + − + + 3 1 3 3 3 1 3 3 1 3 1 3 2 3 1 k k k k k k k k 1 1 Subtract and add 3 .k Because ⋅ 3 2 k is divisible by 2 and − 3 1 k is divisible by 2, it follows that ( ) ⋅ + − = − + 3 2 3 1 3 1 k k k 1 is divisible by 2. Condition II is also satisfied. As a result, the statement “ − 3 1 n is divisible by 2” is true for all natural numbers n. Skill Building 12.5 Assess Your Understanding In Problems 1–22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1. ( ) + + + + = + n n n 2 4 6 2 1 2. ( ) ( ) + + + + − = − n n n 1 5 9 4 3 2 1 3. ( ) ( ) + + + + + = + n n n 3 4 5 2 1 2 5 4. ( ) ( ) + + + + + = + n n n 3 5 7 2 1 2 5. ( ) ( ) + + + + − = + n n n 2 5 8 3 1 1 2 3 1 6. ( ) ( ) + + + + − = − n n n 1 4 7 3 2 1 2 3 1 7. + + + + = − − 1 2 2 2 2 1 n n 2 1 8. ( ) + + + + = − − 1 3 3 3 1 2 3 1 n n 2 1 Proceed as follows: k k k k k k k k k k k k k k 1 2 3 1 1 2 3 1 1 2 1 2 2 2 3 2 2 1 2 2 2 2 ( ) [ ] ( ) ( ) ( ) ( )( ) +++++ +=++++ + + = + + + = + + + = + + = + + Condition II also holds. As a result, formula (4) is true for all natural numbers n. ( ) = + k k 1 2 by equation (5) Now Work PROBLEM 19 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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