This paper proposes a method of using abstract interpretation for discovering properties about array contents in programs which manipulate one-dimensional or multi-dimensional arrays by sequential traversal. The method directly treats invariant properties (including interval universally quantified formulas and atomic formulas) set as abstract domains. It synthesizes invariants by "iterating forward" analysis. This method is sound and converges in finite time. The paper demonstrates the flexibility of the method by some examples. The method has been implemented in a prototype tool. The experiments applying the tool on a variety of array programs (including array-examples benchmark of competition on software verification) demonstrate the feasibility and effectiveness of the approach.