Obtain the sol'n for an=5an-2 - 4an-4
With:
a0 = 3
a1 = 2
a2 = 6
a3 = 8
Note:
The 5an-2 minus 4an-4 actually appear in the the problem as 5a and 4a with the terms being n-2 and n-4 list slightly lower than the an term
$\displaystyle a_n-5a_{n-2}+4a_{n-4}=0$
The characteristic equation is $\displaystyle t^4-5t^2+4=0$ with the roots
$\displaystyle t_1=1, \ t_2=-1, \ t_3=2, \ t_4=-2$
We have to find the general term $\displaystyle a_n$ in the form $\displaystyle a_n=a+b(-1)^n+c\cdot 2^n+d(-2)^n$
For $\displaystyle n=0, \ 1, \ 2, \ 3$ we have
$\displaystyle a_0=a+b+c+d=3$
$\displaystyle a_1=a-b+2c-2d=2$
$\displaystyle a_2=a+b+4c+4d=6$
$\displaystyle a_3=a-b+8c-8d=8$
Now find a, b, c, d.