Discrete Mathematics with Graph Theory (3rd Edition) 117

# Discrete Mathematics with Graph Theory (3rd Edition) 117 -...

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Section 5.1 115 the fonnula is correct. Now suppose that k ~ 1 and the fonnula is correct for n = k. Then k+l k I)2i - 1)(2i) = ~)2i - 1)(2i) + [2(k + 1) - 1)[2(k + 1)] i=l i=l = k(k + 1)(4k - 1) + (2k + 1)(2k + 2) (by the induction hypothesis) 3 k(k + 1)(4k -1) + 6(2k + l)(k + 1) 3 (k + 1)(4k 2 - k + 12k + 6) 3 (k + l)(k + 2)(4k + 3) 3 (k + l)[(k + 1) + 1][4(k + 1) -1] 3 as required. By the Principle of Mathematical Induction, the fonnula holds for all n ~ 1. 9. (a) [BB] If n = 5, 25 = 32, 52 = 25. Since 32 > 25, the result is true for n = 5. Now suppose that k ~ 5 and the result is true for n = k; that is, suppose that 2k > k 2 . We must prove that the result is true for n = k + 1; that is, we must prove that 2k+l > (k + 1)2. Now 2 k + 1 = 2 . 2k > 2k2 by the induction hypothesis, and 2k2 = k 2 + k 2 ~ k 2 + 5k (since k ~ 5), and k 2 +5k = k2+4k+k > k 2
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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 2-3 = 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.

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