Solving a system of coupled recurrence relations

Question

I am required to solve the below system of recurrence relations:$$\begin{cases}a_n-a_{n-4}=t_n-t_{n-3}\\a_n-a_{n-1}=25t_{n-1}-t_{n-3}\end{cases}$$As you can see, I can't isolate $a_i$ or $t_i$. I would like to know if a non-constant closed-form solution exists and if yes, how to obtain it.

Answer 1

Hint:

$$a_n - a_{n-4}=(a_n - a_{n-1}) + (a_{n-1} - a_{n-2}) + (a_{n-2} - a_{n-3}) + (a_{n-3} - a_{n-4})$$

Here, the LHS and each summand on the RHS can be expressed via $t$ using one of your two equations, giving a recurrence relation in just $t$.