SECTION 12.5 Mathematical Induction 895 Solution First show that statement (2) holds for = n 1. Because = 1 1 ,2 statement (2) is true for = n 1. Condition I holds. Next, show that Condition II holds. From statement (2), assume that ( ) + + + + − = k k 1 3 5 2 1 2 (3) is true for some natural number k. Now show that, based on equation (3), statement (2) holds for + k 1. Look at the sum of the first + k 1 positive odd integers to determine whether this sum equals ( ) + k 1 . 2 k k k k k k k k k 1 3 5 2 1 2 1 1 1 3 5 2 1 2 1 2 1 2 1 1 2 2 2 ( ) ( ) [ ] ( ) [ ] ( ) ( ) ( ) ++++ −+ +−=++++ − + + = + + = + + = + Conditions I and II are satisfied; by the Principle of Mathematical Induction, statement (2) is true for all natural numbers n. Using Mathematical Induction Show that the following statement is true for all natural numbers n. > n 2n Solution EXAMPLE 2 First, show that the statement > n 2n holds when = n 1. Because = > 2 2 1, 1 the inequality is true for = n 1. Condition I holds. Next, assume the statement holds for some natural number k; that is, > k 2 . k Now show that the statement holds for + k 1; that is, show that > + + k 2 1. k 1 = ⋅ > ⋅ = + ≥ + + k k k k 2 2 2 2 1 k k 1 ↑ ↑ >k 2k ≥k 1 If > k 2 , k then > + + k 2 1, k 1 so Condition II of the Principle of Mathematical Induction is satisfied. The statement > n 2n is true for all natural numbers n. Using Mathematical Induction Show that the following formula is true for all natural numbers n. ( ) + + + + = + n n n 1 2 3 1 2 (4) Solution EXAMPLE 3 First, show that formula (4) is true when = n 1. Because ( ) + = ⋅ = 1 1 1 2 1 2 2 1 Condition I of the Principle of Mathematical Induction holds. Next, assume that formula (4) holds for some natural number k, and determine whether the formula then holds for the next natural number, + k 1. Assume that ( ) + + + + = + k k k k 1 2 3 1 2 for some (5) Now show that k k k k k k 1 2 3 1 1 1 1 2 1 2 2 ( ) ( ) ( ) [ ] ( )( ) + + + + + + = + + + = + + (continued) =k by equation (3) 2
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