# hw1KEY - COP 3503H Spring 2001 - Homework - Induction...

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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.

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hw1KEY - COP 3503H Spring 2001 - Homework - Induction...

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