Evaluation of function in functional equation

[man111]

If $f(n+2)-5f(n+1)+6f(n)=0$ and $f(0)=2$ and $f(1)=5\;,$ Then $f(n)=$

[nevrodermit]

The characteristics polynomial is
[tex]x^2-5x+6=0[/tex] with discriminant [tex]D=1[/tex] and thus roots [tex]x_{1,2}=\frac{5\pm 1}{2}\Rightarrow x_1=3, x_2=2[/tex].
So [tex]f(n)=c_12^n+c_23^n[/tex].
W are given that [tex]2=f(0)=c_1+c_2[/tex] and [tex]5=f(1)=2c_1+3c_2[/tex] and so [tex]5=2(c_1+c_2)+c_2\Rightarrow c_2=1\Rightarrow c_1=1[/tex] and hence [tex]f(n)=2^n+3^n[/tex].

[man111]

Thanks nevrodermit But i did not understand how can i calculate Characteristic equation.

[nevrodermit]

The theory is described here
https://en.wikipedia.org/wiki/Recurrence_relation

the section name is "Solving homogeneous linear recurrence relations with constant coefficients"