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

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

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