Time Invariance
For a fixed value of $ k $ input to the system $ Y_k[n]=(k+1)^2\delta[n-(k+1)] $, the system is time invariant. As $ n $ goes through a time shift, the output will shift through time an equal amount. Since $ k $ is assumed to be fixed, the output amplitude does not change with 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.
