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Unformatted text preview: Math 23b Homework 4 Solutions 1. Let Q ( n ) be the statement ¬ P ( n ), so that Q ( n ) is true if and only if P ( n ) is false. We will prove that Q ( n ) is true for all n ∈ N by induction. Since P (1) is false, Q (1) is true. Now suppose Q ( n ) is true for some n . Then P ( n ) is false, so by assumption P ( n + 1) is also false, and hence Q ( n + 1) is true. Thus, by induction, Q ( n ) is true for all n , so P ( n ) is false for all n ∈ N . 2. We will prove the statement P ( n ) : ∑ n i =1 i ( i + 1) = n ( n +1)( n +2) 3 for all n ∈ N , by induction. For n = 1, we have ∑ 1 i =1 i ( i + 1) = 1(1 + 1) = 1(1+1)(1+2) 3 . Now suppose ∑ n i =1 i ( i + 1) = n ( n +1)( n +2) 3 . Then ∑ n +1 i =1 i ( i +1) = ∑ n i =1 i ( i +1)+( n +1)( n +2) = n ( n +1)( n +2) 3 + 3( n +1)( n +2) 3 = ( n +1)( n +2)( n +3) 3 . Thus, by induction, P ( n ) is true for all n ∈ N . 3. Let a i ∈ R . We will prove the statement P ( n ) :  ∑ n i =1 a i  ≤ ∑ n i =1  a i  for all n ∈...
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This note was uploaded on 10/15/2011 for the course MATH 23b taught by Professor Bonglian during the Spring '09 term at Brandeis.
 Spring '09
 BongLian
 Math

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