hw2 - x 6 = 1, then x + 2 x 2 + 3 x 3 + ··· + nx n = x-(...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
ECE 103 Discrete Math for Engineers University of Waterloo Spring 2010 Instructors: Koray Karabina, Ashwin Nayak Homework 2, due May 17, 2010 Note: Each question carries five marks. Question 1. Recall that log 2 3 is the real number x such that 3 = 2 x . Prove that log 2 3 is an irrational number. Question 2. What is wrong with the following “proof” that all horses are the same color? Let P ( n ) be the proposition that all the horses in a set of n horses are the same color. Clearly, P (1) is true. Now assume that P ( n ) is true, so that all the horses in any set of n horses are the same color. Consider any n + 1 horses; number these as horses 1 , 2 , 3 ,...,n,n + 1. Now the first n of these horses all must have the same color, and the last n of these must also have the same color. Since the set of the first n horses and the set of the last n horses overlap, all n + 1 must be the same color. This shows that P ( n + 1) is true and finishes the proof by induction. Question 3. Prove that if
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: x 6 = 1, then x + 2 x 2 + 3 x 3 + ··· + nx n = x-( n + 1) x n +1 + nx n +2 (1-x ) 2 for every positive integer n . Question 4. Let H n represent the partial sum of the Harmonic series: H n = 1 + 1 2 + 1 3 + 1 4 + ··· + 1 n . Show that H 2 k ≤ 1 + k for all k ≥ 0. Question 5. In the mound-splitting game, we start off with a single mound of pebbles. In each move, we pick a mound, split it into two smaller mounds of arbitrary size (say, k and m pebbles), multiply the number of pebbles in the two mounds and write down the result (that is, k × m ). We continue playing until every mound has only one pebble (for which we write down the number 0). At the end, we add up all the numbers written down after the splits. Prove that if we start with n pebbles, then the final sum is n ( n-1) / 2 irrespective of how we split the mounds, or in which order we split them . 1...
View Full Document

This note was uploaded on 09/28/2011 for the course ECE 103 taught by Professor Nayak during the Spring '11 term at Waterloo.

Ask a homework question - tutors are online