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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.
- Spring '08