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The Fibonacci sequence F n is defined by F 0 = 0, F 1 = 1, and F n = F n-1 +F n-2 ,
- The Fibonacci sequence Fn is defined by F0 = 0, F1 = 1, and Fn = Fn-1 +Fn-2 , n 2. Show the following.
Fn = (1/sqrt5) ( [(1+sqrt5)/2]n [(1-sqrt5)/2]n ), for all n 0.
Thus, Fn is Q((1/sqrt5) [(1+sqrt5)/2]n ).
If Tn is the number of additions required to compute Fn , then T0 = T1 = 0, and Tn = Tn-1 + Tn-2 + 1, n 2. Hence,
Tn = (1/sqrt5) ( [(1+sqrt5)/2]n+1 [(1-sqrt5)/2]n+1 ) - 1, for all n 0.
Thus, Tn is Q((1/sqrt5) [(1+sqrt5)/2]n+1 ). This is the complexity of the recursive algorithm to compute Fn. Develop a Q(n) non-recursive algorithm to compute Fn .
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