Can the System Time Invariant? Part A
I think that the system is time variant. It seems this way to me because it looks like the time shift is what varies the amplitude of the discrete signal. When there is no time shift, the system has amplitude of 1. For example though, when the time shift increases to three, the amplitude is nine. The system follows a square effect because of the $ k^2+1 $ that each output signal is affected by.
Part B
Assuming the system is linear, the required input would be $ X[n]=u[n] $.
