Discrete Mathematics with Graph Theory (3rd Edition) 160

Discrete Mathematics with Graph Theory (3rd Edition) 160 -...

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158 Solutions to Review Exercises which is indeed true. The induction step is complete; we conclude that (1 - ~) n ;::: 1 - ~ for all n;:::1. 4. When n = 1, the sum is 1 > 2(v'2 - 1) because v'2 ~ 1.4. Now assume that k > 1 and that L~=1 ~ > 2( y'kTI -1). We have k+l 1 k 1 1 2:-=2:-+-= i=1 Vi i=1 Vi y'kTI >2(Vk+l-1)+ ~ yk+1 2(k + 1) - 2y'kTI + 1 y'kTI by the induction hypothesis We want to show that this quantity is at least as large as 2Jk + 2 - 2, in order to get the desired inequality with n = k + 1. Now 2(k + 1) - 2y'kTI + 1 > 2Vk+2 _ 2 Vk+1 - + if and only if 2(k + 1) - 2Vk+1 + 1;::: 2v'k"TIVk"+2 - 2y'kTI if and only if 2(k + 1) + 1 ;::: 2Jk2 + 3k + 2 if and only if 2k + 3;::: 2Jk2 + 3k + 2 and this is true because the numbers on each side of this inequality are positive and (2k + 3)2 4k2 + 12k + 9 > 4(k 2 + 3k + 2). We have established the inequality for n = k + 1 and conclude, by the Principle of Mathematical Induction, that it holds for all n ;::: 1. 5. When
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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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