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Unformatted text preview: COP 3503H Spring 2001  Homework  Induction Proofs SOLUTIONS For each of the following conjectures, produce an induction proof which proves the conjecture is true. Note that all of these conjectures are true. 1. 2200 n 1, and n natural numbers, it is true that = = n 1 i i 2 ) 1 n ( n + Solution Basis : n=1, by definition = = n 1 i i = = 1 1 i i 1, substituting = = 1 1 i i 2 ) 1 1 ( 1 + = 2 2 = 1 So the base case is true! Inductive Hypothesis : n=k, assume = = n 1 i i 2 ) 1 k ( k + is true. Inductive Step : prove conjecture is true for n = k+1 Must prove that: + = = 1 k 1 i i 2 ] 1 ) 1 k )[( 1 k ( + + + = 2 ) 2 k )( 1 k ( + + Note that + = = 1 k 1 i i = k 1 i i + (k +1) = 2 ) 1 k ( k + + (k + 1) Rewriting gives: 2 k k 2 + + k + 1 = 2 k 2 + 2 k + 2 k 2 + 2 2 2 k 2 + 2 k + 2 k 2 + 2 2 = 2 2 k 3 k 2 + + = 2 ) 2 k )( 1 k ( + + proven 2. The harmonic numbers are H 1 , H 2, H 3, , H n where H 1 = 1, H 2 = 1 + , H 3 = 1 + + 1/3 in general H n...
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This note was uploaded on 02/22/2009 for the course COP 3503c taught by Professor Staff during the Spring '08 term at University of Central Florida.
 Spring '08
 Staff
 Computer Science

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