$$a_{n+2} = 9a_{n+1} - 18a_n,\quad n\geq 0,\,\,a_0=1,\,\, a_1=3$$
I got to the point where i moved all to LHS which gives me $a_{n+2} - 9 a_{n+1} + 18 a_n$ (correct me if I'm wrong). I then calculated to the point of getting the roots 3 & 6. From there i don't really know how to get the ''$a_n$''. Could anyone point if i made any mistakes so far and help me get the an?
Thank you
You write the characteristic equation: $r^2=9r-18$ which give you $r_1=3,r_2=6$. The general term has the form $a\cdot 3^n+b\cdot 6^n$. To find $a,b$ you use the relations $a_1=3,a_0=1$ which will give you a $2 \times 2$ system of equations.