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quiz5 - •(1 1 ∈ S • If a,b ∈ S then(2 a 3 b ∈ S...

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Name: Math 55 Quiz 5 SOLUTIONS July 15, 2009 GSI: Rob Bayer You have until 4:00 to complete this quiz. You must show your work. 1. (5 pts) Prove that 1 2 + 2 2 + 3 2 + ··· + n 2 = n ( n +1)(2 n +1) 6 We’ll go by induction on n : BC: If n = 1, then the LHS is 1 2 = 1 and the RHS is 1(1+1)(2+1) 6 = 1 IH: Suppose 1 2 + ··· + k 2 = k ( k +1)(2 k +1) 6 IS: We want to show this works for k + 1 as well: 1 2 + 2 2 + ··· + k 2 + ( k + 1) 2 IH = k ( k + 1)(2 k + 1) 6 + ( k + 1) 2 = k ( k + 1)(2 k + 1) + 6( k + 1) 2 6 = ( k + 1)( k (2 k + 1) + 6( k + 1)) 6 = ( k + 1)(2 k 2 + k + 6 k + 6 6 = ( k + 1)( k + 2)(2 k + 3) 6 as was to be shown. 2. (5 pts) Consider the set S defined recursively as follows:
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Unformatted text preview: • (1 , 1) ∈ S • If ( a,b ) ∈ S , then (2 a, 3 b ) ∈ S Prove that if ( a,b ) ∈ S , then ( a,b ) = (2 n , 3 n ) for some nonnegative integer n . We’ll go by structural induction: • BC: (1 , 1) = (2 , 3 ) • IH: Suppose ( a,b ) ∈ S and ( a,b ) = (2 k , 3 k ). • IS: Then (2 a, 3 b ) = (2 · 2 k , 3 · 3 k ) = (2 k +1 , 3 k +1 ) Thus, by structural induction it works for all ( a,b ) ∈ S...
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