Steps to solve this recurrence relation? I have the following question:


I am aware that I have to find the characteristic polynomial of this equation but I do not understand how to deal with $64 . 3^{n-4}$ so could anyone explain how to solve a recurrence relation of this type?

Also, since this problem expands no $a_{n-4}$ does this mean the characteristic polynomial will be something with $x^4$ ??
After getting the homogeneous solution $a_{h,n}$ from the characteristic equation ($a_{h,n}$ is the solution of the equation $a_{h,n}=6\ a_{h,n-2} + 8\ a_{h,n-3}+ 3\ a_{h,n-4}$), we are just assuming a particular solution is $a_{p,n}=c_n \cdot 3^n$. All we need to do is  apply to the equation and find the $c_n$.

$$ a_{p,n}=6 a_{p,n-2} + 8 a_{p,n-3} + 3 a_{p,n-4} + 64\cdot 3^{n-4} \\ \Longrightarrow c_n\ 3^n=6c_{n-2}3^{n-2}+8c_{n-3}3^{n-3}+3c_{n-4}3^{n-4}+64\cdot3^{n-4} \\ \Longrightarrow c_n= 6c_{n-2}/9+8c_{n-3}/27+3c_{n-4}/81+64/81 $$

If you will get $a_{p,n}$ properly, the original solution is follow

$$a_n=a_{h,n}+a_{p,n}$$

For the entire procedure,


Find a homogeneous solution from the characteristic polynomial. $a_{h,n}$
Do not apply initial conditions because it is not complete. 
Find a particular solution from the above method. $a_{p,n}$
Create a general solution using $a_n=a_{h,n}+a_{p,n}$
Solve the general solution using the initial conditions.
After getting the homogeneous solution $a_{h,n}$ from the characteristic equation ($a_{h,n}$ is the solution of the equation $a_{h,n}=6\ a_{h,n-2} + 8\ a_{h,n-3}+ 3\ a_{h,n-4}$), we are just assuming a particular solution is $a_{p,n}=c_n \cdot 3^n$. All we need to do is  apply to the equation and find the $c_n$.

$$ a_{p,n}=6 a_{p,n-2} + 8 a_{p,n-3} + 3 a_{p,n-4} + 64\cdot 3^{n-4} \\ \Longrightarrow c_n\ 3^n=6c_{n-2}3^{n-2}+8c_{n-3}3^{n-3}+3c_{n-4}3^{n-4}+64\cdot3^{n-4} \\ \Longrightarrow c_n= 6c_{n-2}/9+8c_{n-3}/27+3c_{n-4}/81+64/81 $$

If you will get $a_{p,n}$ properly, the original solution is follow

$$a_n=a_{h,n}+a_{p,n}$$

For the entire procedure,


Find a homogeneous solution from the characteristic polynomial. $a_{h,n}$
Do not apply initial conditions because it is not complete. 
Find a particular solution from the above method. $a_{p,n}$
Create a general solution using $a_n=a_{h,n}+a_{p,n}$
Solve the general solution using the initial conditions.