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MAS111_09_assignment_02

# MAS111_09_assignment_02 - 1 2 n Ā 7 The Fibonacci sequence...

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NANYANG TECHNOLOGICAL UNIVERSITY MAS 111 FOUNDATION OF MATHEMATICS Assignment 2 TUTORIAL TIME:31/08, 1, 2/09, 2009 In question 7, several properties of Fibonacci sequence are listed. Try to find proofs by yourself. After proving these by mathematical induction (you can definitely do like that), try alternative ways - it’s more challenging and more interesting. Have fun! 1. For any positive odd numbers m, n , prove that m 2 + n 2 is even, but not a multiple of 4. 2. Prove that for any natural number n , 3 does not divide n 2 + 1. 3. Prove that 5 + 7 is an irrational number. 4. Prove that for any natural number n , 2 2 n - 1 is divisible by 3. 5. Prove that for any natural number n , 57 divides 7 n +2 + 8 2 n +1 . 6. Prove that 1 - 1 2 + 1 3 - 1 4 - · · · + 1 2 n - 1 - 1 2 n equals to 1 n + 1 + 1 n + 2 +

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Unformatted text preview: + 1 2 n Ā· 7. The Fibonacci sequence is deļ¬ned as follows: f = 0 , f 1 = 1 , f n = f n-1 + f n-2 for n ā„ 2 . The ļ¬rst few Fibonacci numbers are , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , Ā·Ā·Ā· . (a) Prove that for any n ā„ 1, f 1 + f 2 + Ā·Ā·Ā· + f n = f n +2-1. (b) Prove that for any n ā„ 1, f 1 + f 3 + Ā·Ā·Ā· + f 2 n-1 = f 2 n . (c) Prove that for any n ā„ 1, f 2 1 + f 2 2 + Ā·Ā·Ā· + f 2 n = f n f n +1 . (d) Prove that for any n ā„ 1, f n-1 f n +1-f 2 n = (-1) n . This is called Casinniās Identity . (e) Prove that for any n ā„ 1, f n and f n +1 are relatively prime. (f) Suppose that x 2 = x + 1. Prove that for any n ā„ 2, x n = f n x + f n-1 ....
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