Time Invariance
Assuming the subscript $ k $ denotes a time shift in the input signal, $ Y_k[n]=(k+1)^2\delta[n-(k+1)] $ is not time invariant. As the input goes through a time shift, the output amplitude change is related to the square in the shift in time.
Linearity
Since the system is linear, the required input will be $ X[n]=u[n] $. An input of a $ \delta $ functional produces an output of a time-shifted $ \delta $ functional, so an input of a unit step function will produce an output of a time-shifted unit-step function.
