  Let I = (F_1,...,F_r) be a homogeneous ideal of R = k[x_0,...,x_n] generated
by a regular sequence of type (d_1,...,d_r). We give an elementary proof for an
explicit description of the graded Betti numbers of I^s for any s \geq 1. These
numbers depend only upon the type and s. We then use this description to: (1)
write H_{R/I^s}, the Hilbert function of R/I^s, in terms of H_{R/I}; (2) verify
that the k-algebra R/I^s satisfies a conjecture of Herzog-Huneke-Srinivasan;
and (3) obtain information about the numerical invariants associated to sets of
fat points in P^n whose support is a complete intersection or a complete
intersection minus a point.
