h2 - 2 n> n 4 Use induction to prove that every postage...

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CS 241, Sect 001 Homework 2 Dr. David Nassimi Foundations I Due: Week-4, Fri. Sep. 25 Fall 2009 Chapter 2: Proofs and Induction 1. For the domain of real numbers, prove that if ( x < y ) then x < x + y 2 < y. Provide the proof in each of the following ways: (a) Direct Method ; (b) Contrapositive method . 2. Let the average of n real numbers ( x 1 ,x 2 , ··· ,x n ) be A = x 1 + x 2 + ··· + x n n Prove by contradiction that S : i ( x i A ) ∧∃ j ( x j A ) . Hint: In order to prove S is true, start by supposing that S is false, and show that will lead to a contradiction (which cannot be true), thus concluding that S must be true. 3. Use induction to prove each of the following. (a) Arithmetic series sum: n X i =1 ( i ) = n ( n + 1) 2 (b) Geometric series sum, a 6 = 1: n X i =0 a i = a n +1 - 1 a - 1 . (c) All integers of the following form are divisible by 4, for all n 1. f ( n ) = 5 n - 1 (d) For all integers n 1, the following inequality holds:
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Unformatted text preview: 2 n > n. 4. Use induction to prove that every postage of 24 cents or more can be achieved using only 5-cent and 7-cent stamps. That is, prove that for every integer n ≥ 24, there exist some non-negative integers A and B such that P ( n ) : n = 7 A + 5 B. Let P ( n ) be the predicate “postage of n cents can be achieved.” Provide the inductive proof by using each of the following methods. (a) (Simple Induction) Prove the induction base, n = 24. Then, for any n ≥ 24, prove that if P ( n ) is true, then P ( n + 1) will be true. (b) (Strong Induction) Prove the induction base cases (24 , 25 , ··· , 30). Then for n ≥ 31, prove that: If P ( m ) is true for all 24 ≤ m < n , then P ( n ) is also true. 1...
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