  In this paper we give a generalization of a result of Herzog, Hibi, and Zheng
providing an upper bound for regularity of powers of an ideal. As the main
result of the paper, we give a simple criterion in terms of Rees algebra of a
given ideal to show that high enough powers of this ideal have linear
resolution. We apply the criterion to two important ideals $J,J_{1}$ for which
we show that $J^{k},$ and $J_{1}^{k}$ have linear resolution if and only if
$k\neq 2.$ The procedures we include in this work is encoded in computer
algebra package CoCoA.
