Computing ϕ = (1 + √ 5)/2. a.) Let Fn denote the n th Fibonacci number, which are generated by the recursion F0 = F1 = 1 , Fn+1 = Fn + Fn−1 . You may assume that limn→∞ Fn+1 Fn = ϕ = 1 + √ 5 2 . b.) Use the ratios of Fibonacci numbers to compute ϕ. Note that you are using the continued fraction representation for ϕ =  to approximate ϕ. List your output as in the following table. Iteration Number Approximation Number of Correct Digits c.) How many steps does your algorithm take to approximate ϕ to 10 digits? 15 digits? 20 digits? 30 digits? d.) Use the recursion xn+1 = xn − (x 2 n − xn − 1) (2xn − 1) , x0 = 2 To compute approximations of ϕ. As you know from class, this recursion is developed from the Newton-Raphson method applied to f(x) = x 2 − x − 1, with a starting value of x0 = 2. List your output as in the table above.

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