The Fibonacci sequence, first studied by the thirteenthcentury Italian mathematician
Chapter 9, Problem 59(choose chapter or problem)
The Fibonacci sequence, first studied by the thirteenthcentury Italian mathematician Leonardo di Pisa, alsoknown as Fibonacci, is defined recursively byFn = Fn1 + Fn2 for n > 2 and F1 = 1, F2 = 1.The Fibonacci sequence occurs in many branches ofmathematics and can be found in patterns of plant growth(phyllotaxis).(a) Find the first 12 terms.(b) Show that the sequence of successive ratiosFn+1/Fn appears to converge to a number r satisfyingthe equation r2 = r + 1. (The number r wasknown as the golden ratio to the ancient Greeks.)(c) Let r satisfy r2 = r + 1. Show that the sequencesn = Arn, where A is constant, satisfies the Fibonacciequation sn = sn1 + sn2 for n > 2.
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