23b_hw4_solutions

23b_hw4_solutions - Math 23b Homework 4 Solutions 1 Let Q(n...

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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 N , by induction. For n = 1, we have 1 i =1 a i = | a 1 | = 1 i =1 | a i | . Now fix n N and suppose P ( n ) is true. Note that, by the triangle inequality, we have n +1 X i =1 a i = n X i =1 a i + a n +1 n X i =1 a i + | a n +1 | (1)
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