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# pmi09f - x and y so that n = 2 x 5 y Hint(2 x 5 y 2 = 2 x 1...

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PRINCIPLE OF MATHEMATICAL INDUCTION (MI) Let N = { 1 , 2 , 3 , . . . } be the natural numbers. Let Z = { . . . , - 2 , - 1 , 0 , 1 , 2 , . . . } be the integers. Let P ( n ) be a statement (that is either true or false) about n . Theorem 1.2.1: PMI (basic form) If base step: P (1) is true inductive step: for each n N : [ P ( n ) is true ] = [ P ( n + 1) is true ] then P ( n ) is true for each n N . The proof of the PMI is based Peano’s Postulates of N . Theorem 1.2.2: PMI (doesn’t matter where you start form) Fix n 0 Z . If base step: P ( n 0 ) is true inductive step: for each n Z with n n 0 : [ P ( n ) is true ] = [ P ( n + 1) is true ] then P ( n ) is true for each n Z such that n n 0 . Theorem 1.2.3: PMI (strong form) Fix n 0 Z . If base step: P ( n 0 ) is true inductive step: for each n Z with n n 0 : [ P ( j ) is true for j = n 0 , 1 + n 0 , . . . , n ] = [ P ( n + 1) is true ] then P ( n ) is true for each n Z such that n n 0 . 1

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Example 1: Prove that n 3 < n ! for all integers n 6 Example 2: Show that each natural number greater than 3 can be written as a linear comination of the numbers 2 and 5; that is, for all
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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 deﬁned 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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