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Unformatted text preview: + 1 2 n · 7. The Fibonacci sequence is deﬁned as follows: f = 0 , f 1 = 1 , f n = f n1 + f n2 for n ≥ 2 . The ﬁrst few Fibonacci numbers are , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , ··· . (a) Prove that for any n ≥ 1, f 1 + f 2 + ··· + f n = f n +21. (b) Prove that for any n ≥ 1, f 1 + f 3 + ··· + f 2 n1 = f 2 n . (c) Prove that for any n ≥ 1, f 2 1 + f 2 2 + ··· + f 2 n = f n f n +1 . (d) Prove that for any n ≥ 1, f n1 f n +1f 2 n = (1) n . This is called Casinni’s Identity . (e) Prove that for any n ≥ 1, f n and f n +1 are relatively prime. (f) Suppose that x 2 = x + 1. Prove that for any n ≥ 2, x n = f n x + f n1 ....
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This note was uploaded on 01/23/2011 for the course MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.
 Spring '10
 DrChanSongHeng

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