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A closed-form expression for the Fibonacci numbers

A closed-form expression for the Fibonacci numbers

A closed-form expression for the Fibonacci

A closed-form expression for the Fibonacci numbersK. ShumMar, 2010.Defintion of Fibonacci numbers:Let F 0 =0, F 1 = 1, and define F n recursively by F n = F n-1 + F n-2 , for n 2.The first few Fibonacci numbers are: 0,1,1,2,3,5,8,13,21,…Objective: find a closed-form expression of F n , and prove thatlim→F = 1 + √5F 2= the Golden ratio.‣ Trick #1: Write the recurrent relation as F F = 1 11 0 F F .The first equation is the defining equation for F t+1 : F t+1 = F t + F t-1 . Interpret F F as the state of a dynamical system at time t. To calculate the stateat time t+1, we multiply the state at time t by 1 1 . We have the1 0following key equation F = 1 1F 1 0 F = 1 1F 1 0 1 . (*)0The calculation is now reduced to the computation of 1 11 0 .‣ Compute the eigenvalues of 1 1 . The characteristic equation is1 0After some simplifications, we get 1 − λ 11 −λ = 0.λ − λ − 1 = 0.The roots of this quadratic polynomial are1

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