The goal in this problem is to determine the generating function W(x) for the counting sequence of the family W of binary strings with the special property that every 1 is followed by an odd number of 0's. (So, if W(x)--0wnz", then wn is the number of strings in W of length n) (a) Show that the generating function W%,1 for the family Ws,1 of strings composed of a single 1 followed by an odd number of zeros satisfies 2 (b) Show that the generating function W, for the family Ws of strings in W that start with a 1 satisfies (c) Use the rational expression for 12 + together with (a) and (b) and an infinite substitution to show that 1 - 2r2 (d) Use (c) to show that 1 - 2x The goal in this problem is to determine the generating function W(x) for the counting sequence of the family W of binary strings with the special property that every 1 is followed by an odd number of 0's. (So, if W(x)--0wnz", then wn is the number of strings in W of length n) (a) Show that the generating function W%,1 for the family Ws,1 of strings composed of a single 1 followed by an odd number of zeros satisfies 2 (b) Show that the generating function W, for the family Ws of strings in W that start with a 1 satisfies (c) Use the rational expression for 12 + together with (a) and (b) and an infinite substitution to show that 1 - 2r2 (d) Use (c) to show that 1 - 2x