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Solution.The Fibonacci numbers are defined by the recurrence relation is defined F1 = 1, F2 =...

Solution.The Fibonacci numbers are defined by the recurrence relation is defined F1 = 1, F2 = 1 and for n > 1, Fn+1 = Fn + Fn−1. So the first few Fibonacci Numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . There are numerous curious properties of the Fibonacci Numbers Use the method of mathematical induction to verify a: For all integers n > 1 and m > 0 Fn−1Fm + FnFm+1 = Fn+m. Solution. (Hint: use induction with respect to m. First verify the formula for m = 1 case (here use again induction on n) then assume that it is true for m = 1, m = 2, · · · , m = k case and prove that it remains true if m = k + 1 (with the use of induction on n) b: What one can obtain from formula a) after substitution n = m + 1 Solution. c) Use the principle of Mathematical Induction to prove that for n 3 − n is multiple of 3 for all positive integers n.

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in part c I have shown that n^3-n is divisible by 3 for all n ,natural numbers .So n^3-n is multiple of 3.

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