Linearity and Time Variance
From the given equation:
$ \,\!Y_k[n]=(k+1)^2\delta [n-(k+1)] $ ,
It is clear that this is a non-time-invariant, or more simply, a time variant function because the amplitude changing $ (k+1)^2 $ term.
The inputs to this system are all shifted $ \delta $ functions, this means that if the $ X[n] $ inputs were changed to unit step functions $ u[n] $ , then the output will be a time shifted step function. For this specific example we want an output of $ Y[n]=u[n-1] $ , so we can notice that when :
$ \,\!X[n]=\delta [n-1] $ yields $ \,\!Y_0[n]=\delta [n-1] $ , then
$ \,\!X[n]=u[n-1] $ will yield $ \,\!Y_0[n]=u[n-1] $
