Please be detailed and clear

Let A be the set of binary strings containing exactly one 1 . Let Sn={(,):,A,+=n}, where denotes the length of . In A1Q2 we proved that Sn=(n+13). In this question we obtain Sn by the method of generating series. Let be the weight function that returns the length of a binary string. (a) [3 marks] Write down the generating series A(x) for A with respect to weight , in closed form (i.e. without using or ++ ). (b) [3 marks] Let (x)=n0Snxn. Prove that (x)=A(x)2. (c) [3 marks] Using part (b), prove that Sn=(n+13)