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Unformatted text preview: ~ 1),andk 2 +2k+1 = (k+1)2. By the Principle of Mathematical Induction, the result is true for all n ~ 5. (b) Since 23 = l, the inequality holds when n = 3. Now assume that k ~ 3 and 2k ~ l. We have 2k+l = 2(2k) ~ 2(~) 1 1 ="4 > 8' by the induction hypothesis which is the desired result with n = k + 1. By the Principle of Mathematical Induction, we conclude that 2 n > l for all n ~ 3. (c) [BB] Since 6! = 720 > 216 = 6 3 , the inequality holds for n = 6. Now assume that k ~ 6 and that k! > k 3 . We have (k + I)! = (k + l)k! > (k + 1)k 3 = k4 + k 3 ~ 6k 3 + k 3 = k 3 + 3k 3 + 3k 3 ~ k 3 + 3k 2 + 3(36)k ~ k 3 + 3k 2 + 3k + 1 = (k + 1)3 using the induction hypothesis since k ~ 6 since k 2 ~ 36 which is the desired inequality with n = k + 1. By the Principle of Mathematical Induction, we conclude that n! > n 3 for all n ~ 6....
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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