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Unformatted text preview: k − 1)! = = = k (k + 1)(k + 2) (k − 1)! (k − 1)! (k$$$ − 1)! $ The stipulation k ≥ 1 is there to ensure that all of the factorials involved are defined. 2. We proceed by induction and let P (n) be the inequality n! > 3n . The base case here is n = 7 and we see that 7! = 5040 is larger than 37 = 2187, so P (7) is true. Next, we assume that P (k ) is true, that is, we assume k ! > 3k and attempt to show P (k + 1) follows. Using the properties of the factorial, we have (k + 1)! = (k + 1)k ! and since k ! > 3k , we have (k + 1)! > (k + 1)3k . Since k ≥ 7, k + 1 ≥ 8, so (k + 1)3k ≥ 8 · 3k > 3 · 3k = 3k+1 . Putting all of this together, we have (k + 1)! = (k + 1)k ! > (k + 1)3k > 3k+1 which shows P (k + 1) is true. By the Principle of Mathematical Induction, we have n! > 3n for all n ≥ 7. Of all of the mathematical animals we have discussed in the text, factorials grow most quickly. In problem 2 of Example 9.4.1, we proved that n! overtakes 3n at n = 7. ‘Overtakes’ may be too polite a word,...
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