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Unformatted text preview: CSE 21: Solutions  Problem Set 7 1. Proof. Initial step: For n = 1, the lefthand size is 1 3 = 1 and the righthand side is ( 1+1 2 ) 2 = 1. Then the equality holds. Inductive Hypothesis: Suppose it is true for n = t , i.e., 1 3 + 2 3 + + t 3 = t + 1 2 2 . We prove that its true for n = t + 1, i.e., 1 3 + 2 3 + + t 3 + ( t + 1) 3 = t + 2 2 2 . Inductive step: 1 3 + 2 3 + + t 3 + ( t + 1) 3 = ( 1 3 + 2 3 + + t 3 ) + ( t + 1) 3 = t + 1 2 2 + ( t + 1) 3 Inductive hypothesis = ( t + 1) t 2 2 + ( t + 1) 3 = ( t + 1) 2 4 ( t 2 + 4( t + 1) ) = ( t + 1) 2 4 ( t + 2) 2 = ( t + 2)( t + 1) 2 2 = t + 2 2 2 2. Initial step: For n = 1, the lefthand size is 1 2 1 2 0 = 1, and the righthand side is ( 1) 2 = 1. So the equality holds. Inductive hypothesis: Suppose its true for n = k , i.e., F 2 k F k +1 F k 1 = ( 1) k +1 . We prove that its true for n = k + 1, i.e., F 2 k +1 F k +2 F k = ( 1) k +2 ....
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 Spring '07
 Graham

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