  Starting from \cite{Ayy2} we compute the Groebner basis for the defining
ideal, P, of the monomial curves that correspond to arithmetic sequences, and
then give an elegant description of the generators of powers of the initial
ideal of P, inP. The first result of this paper introduces a procedure for
generating infinite families of Ratliff-Rush ideals, in polynomial rings with
multivariables, from a Ratliff-Rush ideal in polynomial rings with two
variables. The second result is to prove that all powers of inP are
Ratliff-Rush. The proof is through applying the first result of this paper
combined with Corollary (12) in \cite{Ayy4}. This generalizes the work of
\cite{Ayy1} (or \cite{Ayy11}) for the case of arithmetic sequences. Finally, we
apply the main result of \cite{Ayy3} to give the necessary and sufficient
conditions for the integral closedness of any power of inP.
