if $x[n] = (-1)^n$ and $y[n] = (-2)^n$, why the system is not LTI?

Question

I am a student and I answered this question "it can be LTI" but my answer was incorrect and the right answer is that this system can't be LTI (Linear time-invariant system). I don't understand why. could you help me? the question is this:

If in a system with the input $x[n] = (-1)^n$ we have the answer $y[n] = (-2)^n$, can this system be LTI?

Why this system can't be LTI?

Answer 1

For time-invariant systems, if the input is periodic, then the output is periodic.

so now if we have an LTI system it means it is time-invariant. and the input is periodic but not the output.

so it can't be LTI.