How to find the general solution of the difference equation

Question

Take this as an example. Wondering if someone can elaborate their ideas. Thanks a lot!

Answer 1

The characteristic polynomial for the given recurrence is: $$ x^2-x-1=0 $$ with distinct, real roots $x_{1,2}=\frac{1\pm\sqrt{5}}{2}$

The general solution to the given recurrence is: $$ \psi=c_1x_1^n+c_2x_2^n $$