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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 n1 + f n2 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 +21. (b) Prove that for any n ā„ 1, f 1 + f 3 + Ā·Ā·Ā· + f 2 n1 = 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 n1 f n +1f 2 n = (1) n . This is called Casinniās 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 n1 ....
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 Spring '10
 DrChanSongHeng
 Natural number, Prime number, Nanyang Technological University, Golden ratio

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