The sequence {Fn} described by F0 = 1, F1 = 1, and Fn+2 = Fn+Fn+1, if n ≥ 0, is called a Fibonacci sequence. Its terms occur naturally in many botanical species, particularly those with petals or scales arranged in the form of a logarithmic spiral. Consider the sequence {xn}, where xn = Fn+1/Fn.
Assuming that limn→∞ xn = x exists, show that x = (1 +√5)/2. This number is called the golden ratio.