Part A
We are given the following: $ X_k[n]=\delta[n-k] \rightarrow \text{ system } \rightarrow Y_k[n]=(k+1)2 \delta[n-(k+1)] \ (k \in \mathbb{Z}, k \geq 0) $
Translate this into math: (See this for symbology.)
Let $ x=\delta[n] $. We are given that the system f works in the following way:
$ f(S_k(x)) = f(S_k(\delta[n])) = f(\delta[n-k]) = (k+1)^2 \delta[n-(k+1)] $
To show whether f is time invariant or time variant, we must examine the following:
$ S_k(f(x)) = S_k(f(\delta[n])) = S_k(\delta[n-1]) = \delta[n-(k+1)] $
Since $ \exists (k \geq 0) \in \mathbb{Z} \ s.t. \ S_k(f(x)) \neq f(S_k(x)) $ (e.g., if k=1), f (ie, the "system") is time variant.
Part B
- coming soon...
