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Unformatted text preview: : T ( n 1) = 1 = 1(1 + 1) / 2 . Induction hypothesis : T ( n 1) = ( n 1)( n ) / 2 . Induction step : T ( n ) = T ( n 1) + n = ( n 1)( n ) / 2 + n = n ( n + 1) / 2 . Thus, the theorem is proved by mathematical induction. 2.28 If we expand the recurrence, we get T ( n ) = 2 T ( n 1) + 1 = 2(2 T ( n 2) + 1) + 1) = 4 T ( n 2 + 2 + 1 . Expanding again yields T ( n ) = 8 T ( n 3) + 4 + 2 + 1 . From this, we can deduce a pattern and hypothesize that the recurrence is equivalent to T ( n ) = n X i =0 12 i = 2 n 1 . To prove this formula is in fact the proper closed form solution, we use mathematical induction. Base case : T (1) = 2 1 1 = 1 ....
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This note was uploaded on 12/27/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.
 Fall '08
 BELL,D

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