Asymptotic behaviour of Castelnuovo-Mumford regularity

Let S be a polynomial ring over a field. For a graded S-module generated in degree at most P, the Castelnuovo-Mumford regularity of each of (i) its n(th) symmetric power, (ii) its n(th) torsion-free symmetric power and (iii) the integral closure of its n(th) torsion-free symmetric power is bounded above by a linear function in n with leading coefficient at most P. For a graded ideal I of S, the regularity of I-n is given by a linear function of n for all sufficiently large n. The leading coefficient of this function is identified.