# 2 - YORK UNIVERSITY Faculty of Arts Faculty of Pure and...

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Unformatted text preview: YORK UNIVERSITY Faculty of Arts Faculty of Pure and Applied Science January - April 2003 AS/SC/MATH 2320 3.0 M Term Test 2b SOLUTIONS 1. (7 marks) Let f 1 = 1 , f 2 = 1 and f n = f n- 1 + f n- 2 , for n = 3 , 4 , 5 , ··· . Show that f n ≤ 2 n , for every positive integer n. Hint: Use the second principle of Mathematical Induction. Answer: Prove: ( ∀ n ∈ Z + ) P ( n ) , where P ( n ) is the proposition f n ≤ 2 n . Basis Step: P (1) is true, since f 1 = 1 and 1 ≤ 2 = 2 1 ; P (2) is also true, since f 2 = 1 and 1 ≤ 4 = 2 2 . Induction Hypothesis: Assume that P ( j ) is true for all j : 1 ≤ j ≤ k, i.e. f j ≤ 2 j , j = 1 ,k. Induction Step: Then P ( k + 1) must be true also. f k +1 = f k + f k- 1 ≤ 2 k + 2 k- 1 (by the induction hypothesis) ≤ 2 k + 2 k = 2 · 2 k = 2 k +1 . Hence, P ( k + 1) is also true. 2. (5 marks) Write a pseudocode description for a recursive algorithm that computes the value of 5 2 n , where n is a nonnegative integer....
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2 - YORK UNIVERSITY Faculty of Arts Faculty of Pure and...

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