By Abraham P Hillman
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2 (a) Fn%1 & Fn Fn%2. 2 (b) 2 Fn%2 & Fn%1 Fn . (c) Fn-1 + Fn+1. 22. Discover and prove formulas similar to the first two parts of the previous problem for the Lucas numbers. 23. Prove the following for all integers m and n: (a) Lm%n%1 ' Fm%1Ln%1 % FmLn. (b) Fm%n%1 ' Fm%1Fn%1 & FmFn. 37 24. Prove that (Fn+1)2 + (Fn)2 = F2n+1 for all integers n. 25. Let a and b be the roots of the quadratic equation x2 - x - 1 = 0. Prove that: (a) Fn ' an & bn . a &b (b) Ln ' a n % b n. (c) FnLn ' F2n. (d) a n ' aFn % Fn&1 and b n ' bF n % Fn&1.
The second method, induction, involves the guessing or discovery of general patterns from observed data. While in most branches of science and mathematics the guesses based on induction may remain merely conjectures, with varying degrees of probability of correctness, certain conjectures in mathematics which involve the integers frequently can be proved by a technique of Pascal called mathematical induction. Actually, this technique in not induction, but is rather an aid in proving conjectures arrived at by induction.
Permutations of 1, 2, 3 are even and three are odd. The even ones are listed in the first column, and the odd ones in the second column: 54 1, 2, 3 1, 3, 2 2, 3, 1 2, 1, 3 3, 1, 2 3, 2, 1 For general n, a permutation i, j, h, k, ... , r, s of 1, 2, 3, ... , n is associated with the product p = [(j - i)][(h - i)(h - j)][(k - i)(k - j)(k - h)] ... (s - r)] of all the differences of two of i, j, h, k, ... , r, s in which the number that appears first is subtracted from the other. If the permutation i, j, h, k, ...