Unformatted text preview: P m ,P m +1 ,... of propositions is true provided (i) P m is true, (ii) P n +1 is true whenever P n is true and n ≥ m . (a) Prove that n 2 > n + 1 for all integers n ≥ 2. For a natural number n let P n denote the inequality n 2 > n + 1. Now the base case is m = 2, so we must verify P 2 . For n = 2, n 2 = 2 2 = 4, while n + 1 = 2 + 1 = 3: since 4 > 3, P 2 holds. Now suppose that P n holds for some integer n ≥ 2. We must check that P n +1 also holds: ( n + 1) 2 = n 2 + 2 n + 1 > ( n + 1) + 2 n + 1 by P n = 3 n + 2 > n + 2 since n ≥ 2 > . Hence, ( n + 1) 2 > ( n + 1) + 1. Thus for every integer n ≥ 2, P n implies P n +1 . By the principle of mathematical induction we conclude that P n is true for every integer n ≥ 2. ±...
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 Spring '10
 janeday
 Mathematical Induction, Natural number, pn

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