hw7 - summation, then convert to a simple algebraic...

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CS 173: Discrete Structures, Spring 2010 Homework 7 This homework contains 4 problems worth a total of 40 regular points. It is due on Friday, March 19th at 4pm. Put your homework in the appropriate dropbox in the Siebel basement. 1. Big-O proofs [10 points] For both parts of this problem, make sure your proof is presented clearly, in forward order, and spells out the (maybe 4-5 steps of) algebra clearly. (a) Prove that 3 x is O ( x 2 x +2 ). (b) Use a proof by contradiction to show that 2 x 3 x +1 is not O ( x ). 2. Unrolling [10 points] Use unrolling to ±nd a closed form for the following functions with domain N and co- domain R . (a) f ( n ) = b f ( n - 7) + 4 if n 7 4 otherwise (b) g ( n ) = b g ( n 3 ) + 21 if n > 1 0 otherwise Show at least three steps of unrolling, show the unrolling pattern compactly using a
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Unformatted text preview: summation, then convert to a simple algebraic equation. For part (b), assume that n is a power of 3. 3. Strong induction [10 points] Suppose that f : N Z is dened by f (0) = 0 f (1) = f (2) = 1 f ( n ) = 2 f ( n-1) + f ( n-2)-2 f ( n-3) for all n 3. Use strong induction to show that f ( n ) = 2 n +(-1) n +1 3 for every natural number n . Hint: you must use strong induction, because thats the main point of this problem. 4. Induction with an inequality [10 points] Consider the sequence given by: a n +1 = 3-1 a n a 1 = 1 Prove by induction that 1 (a) The sequence is increasing, ie: n,a n +1 a n . (b) n,a n < 3. 2...
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This note was uploaded on 11/09/2011 for the course CS 173 taught by Professor Erickson during the Spring '08 term at University of Illinois, Urbana Champaign.

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hw7 - summation, then convert to a simple algebraic...

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