The DT system has an input-output relationship such that when input X[n] = d[n-k] then the output is Y[n] = (k+1)^2d[n-(k+1)] for k >= 0.
(a) This system cannot be time-invariant because, assuming that the variable 'k' represents an amount of time, the computation of the output is based entirely on time with the 'k' variable present in all components of the output function.
(b) Assuming that the system is linear, an input X[n] that would yield the output Y[n]=u[n-1] would be the input X[n] = u[n], a unit step function input that would thus yield a unit step function output.
