Sheet8 - McMaster University Department of Computing and...

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Department of Computing and Software Dr. W. Kahl COMP SCI 1FC3 Exercise Sheet 8 COMP SCI 1FC3 — Mathematics for Computing 13 March 2009 Exercise 8.1 — Mathematical Induction Proofs (Textbook pp. 279–280) Prove the following by mathematical induction: (a) 2 1 + 2 3 + 2 5 + ⋅ ⋅ ⋅ + (2 n + 2 1) = ( n + 1)(2 n + 1)(2 n + 3) / 3 whenever n is a nonnegative integer (b) 1 ⋅ 1! + 2 ⋅ 2! + ⋅ ⋅ ⋅ + n n ! = ( n + 1)! - 1 whenever n is a nonnegative integer (c) n k =1 k k 2 = ( n - n +1 1)2 + 2 whenever n is a positive integer (d) n ! < n n whenever n is an integer greater than 1 Exercise 8.2 — Haskell Evaluation Consider the following Haskell source Evaluation.hs (available on the course page): f 0 = 1 f n = n * f ( n - 1 ) g a 0 = a g a b = if a < b then g b a else g ( a - b ) b h 1 = 1 h n = if even n then h ( n ‘div‘ 2 ) else h ( 3 * n + 1 ) Evaluate manually the following Haskell expressions: (a) f 2 (b) f 4 (c) g 12 6 (d) g 14 10 (e) h 1 (f)
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This note was uploaded on 03/26/2011 for the course CS 1fc3 taught by Professor Kahl during the Spring '11 term at McMaster University.

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Sheet8 - McMaster University Department of Computing and...

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