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**Unformatted text preview: **x and y so that n = 2 x + 5 y . Hint: (2 x + 5 y ) + 2 = 2( x + 1) + 5 y . Reference: [A] An Introduction to Fibonacci Discovery by Brother U. Alfred The Fibonacci Numbers { F n } n =1 The earliest study of these famous numbers is attributed to Leonardo Of Pisa (alias Fibonacci) early in the thirteenth century. They are associated with a variety of natural phenomena. They are dened recursively by F 1 = 1 F 2 = 1 F n = F n-1 + F n-2 for n 3 . So { F n } n =1 = { 1 , 1 , 2 , 3 , 5 , 8 ,... } Example 3 [A, page 9]: Show that for each n 2, F 2 n-F n-1 F n +1 = (-1) n-1 . Example 4 [A, page 13]: Show that for each n N F n = 1 5 " 1 + 5 2 ! n-1- 5 2 ! n # . Notation suggestion: let r = 1 + 5 2 and s = 1- 5 2 . 2...

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