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A closed-form expression for the Fibonacci numbers

A **closed**-**for**m**expression** **for** **the** **Fibonacci** **numbers**K. 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**-**for**m**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 **the**1 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

- Page 2 and 3: λ = 1 + √5 ≈ 1.618, and λ2 =
- Page 4: y rounding√ √ to the nearest