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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Page1 / 4

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online