#2. Fibonacci Numbers, the Formula

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Prompts for discussion:

Exercise 1. Show that: \[ F_n=\left\lfloor\frac{1}{\sqrt{5}} \cdot\left(\frac{1+\sqrt{5}}{2}\right)^n\right\rfloor \]

(Source: generatingfunctionology, Wilf; h/t Matthew Drescher and John Azariah for a fun Twitter discussion on this.)

Exercise 2. Use this method to work out a closed form for:

\[ y_{n+2}=2 y_{n+1}-y_n \]

(Source: This is an exercise from the book.)