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Discrete Mathematics with Graph Theory (3rd Edition) 159

# Discrete Mathematics with Graph Theory (3rd Edition) 159 -...

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Chapter 5 157 Chapter 5 Review 1. When n = 1, the sum is 1(1!) = 1, while the right hand side is 2! - 1 = 1, so the result holds. Now k assume that k 2: 1 and that the result is true for n = k; that is, assume L i(i!) = (k + I)! - 1. We i=l k+1 must prove that L i(i!) = (k + 2)! - 1. We have i=l k+1 k L i(i!) = [L i(i!)] + (k + l)(k + I)! i=l i=l = [(k + I)! - 1J + (k + l)(k + I)! = (k + 1)!(1 + k + 1) - 1 by the induction hypothesis = (k + l)!(k + 2) - 1 = (k + 2)! - 1 as desired. So the result is true, by the Principle of Mathematical Induction. n . 3n -1 3 1 -1 2. For n = 1, L 3,-1 = = 1 while -- = -2- = 1, so the result holds. Now suppose k 2: 1 i=l 2 k k and the result is true for n = k; that is, assume that ~ 3 i - 1 = 3 ; 1.
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