Data Str & Algorithm HW Solutions 10

Data Str & Algorithm HW Solutions 10 - 3 F ( n ) − F...

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10 Chap. 2 Mathematical Preliminaries (c) Induction Step. n X i =1 1 2 i = n 1 X i =1 1 2 i + 1 2 n =1 1 2 n 1 + 1 2 n =1 1 2 n . Thus, the theorem is proved by mathematical induction. 2.20 Proof : (a) Base case. For n =0 , 2 0 =2 1 1=1 . Thus, the formula is correct for the base case. (b) Induction Hypothesis. n 1 X i =0 2 i =2 n 1 . (c) Induction Step. n X i =0 2 i = n 1 X i =0 2 i +2 n =2 n 1+2 n =2 n +1 1 . Thus, the theorem is proved by mathematical induction. 2.21 The closed form solution is 3 n +1 3 2 , which I deduced by noting that
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Unformatted text preview: 3 F ( n ) − F ( n ) = 2 F ( n ) = 3 n +1 − 3 . Now, to verify that this is correct, use mathe-matical induction as follows. For the base case, F (1) = 3 = 3 2 − 3 2 . The induction hypothesis is that ∑ n − 1 i =1 = (3 n − 3) / 2 . So, n X i =1 3 i = n − 1 X i =1 3 i + 3 n = 3 n − 3 2 + 3 n = 3 n +1 − 3 2 . Thus, the theorem is proved by mathematical induction....
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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.

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