Induction Problem with Fibonacci Numbers
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The Fibonacci recurrence is F(0) = F(1) = 1, and F(n) = F(n-1) + F(n-2), for n > 1
The values F(0), F(1), F(2), ... form the sequence of Fibonacci numbers, in which each number after the first two is the sum of the two previous numbers. Let r = (1+ /2). The constant r is called the golden ratio and its value is about 1.62. Show that F(n) is O(r ).
Hint: For the induction, it helps to guess that F(n) ar for some n, and attempt to prove that inequality by induction on n. The basis must incorporate the two values n = 0 and n = 1. In the inductive step, it helps to notice that r satisfies the equation r = r + 1.
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The expert examines Fibonacci Numbers recurrence.
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The Fibonacci recurrence is F(0) = F(1) = 1, and F(n) = F(n-1) + F(n-2), for n > 1
The values F(0), F(1), F(2), ... form the sequence of Fibonacci numbers, in which each number after the first two is the sum of the two previous ...
Purchase this Solution
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