Is this an infinite response or a finite response?
A finite impulse response (FIR) filter has an impulse response of finite length. It can be implemented by a non-recursive structure, but there are also cases where the designer would choose a recursive structure for implementing an FIR filter (e.g., a recursively implemented moving average filter, as explained in this answer).

An infinite impulse response (IIR) filter has an impulse response of infinite extension. Consequently, it cannot be implemented by a non-recursive structure, but it requires a recursive structure.

With the above explanation in mind, I think it should be straightforward to decide which class the system in your question belongs to.