Derive the closed form of the Fibonacci sequence. The Fibonacci sequence is defi

Question

Derive the closed form of the Fibonacci sequence. The Fibonacci sequence is defined recursively as: f(n) = f(n - l) + f(n - 2) f(0) = 0 f(l) = 1 Find the values for the characteristic equation, c_1, x_1, C_2 and x_2 IMPORTANT: for squareroot s_1 enter the answer as follows: squareroot 3 = squareroot (3) Also, if the quadratic equation is used, select the + sign first, then the - sign Characteristic Equation: ____ x^2 + ______ x + _______ = 0 x_1 = _______ (+ part of quadratic equation) x_2 = ________ (- part of quadratic equation) Given f(n) = c_1x_1^n + c_2x_2^n f(0) = c_1 + c_2 f(1) = c_1x_1 + c_2x_2 c_1 = _________ c_2 = _______ The final closed form equation would be:

Explanation / Answer

characteristic equation :x2-x-1 : 1 x2 + -1 x + -1

x1 = (1+sqrt(5))/2

x2 = (1-sqrt(5))/2

c1=1/sqrt(5)

c2= -1/sqrt(5)

f(n) =1/sqrt(5) * ((1+sqrt(5))/2)n + -1/sqrt(5) * ((1-sqrt(5))/2)n   

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