From a GF(q) sequence {ai}i0 with period qn - 1 we can obtain new periodic sequences {ai}i0 with period qn by inserting one symbol b GF(q) at the end of each period. Let b0 = Σqn-2 i=0 ai. It Is first shown that the linear complexity of {ai}i0, denoted as LC({ai}) satisfies LC({ai}) = qn if b -b0 and LC({ai}) qn - 1 if b = -b0 Most of known sequences are shown to satisfy the zero sum property, i.e., b0 = 0. For such sequences satisfying b0 = 0 it is shown that qn - LC({ai}) LC({ai}) qn - 1 if b = 0.