Some problems concerning the Frobenius number for extensions of an arithmetic progression

For positive and relative prime set of integers A={a_1,… ,a_k} , let (A) denote the set of integers of the form a_1x_1+⋯ +a_kx_k with each x_i ≥ 0 . It is well known that ^c(A)=ℕ∖(A) is a finite set, so that (A) , which denotes the largest integer in ^c(A) , is well defined. Let A=AP(a,d,k) denote the set {a,a+d,… ,a+(k-1)d} of integers in arithmetic progression, and let (a,d)=1 . We (i) determine the set A^+={ b ∈^c(A): (A ∪{b})=(A) } ; (ii) determine a subset A^+ of ^c(A) of largest cardinality such that A ∪A^+ is an independent set and (A ∪ A^+)=(A) ; and (iii) determine (A ∪{b}) for some class of values of b that includes results of some recent work.