Generalized asymptotic Euler's relation for certain families of polytopes

According to Euler's relation any polytope P has as many faces of even dimension as it has faces of odd dimension. As a generalization of this fact one can compare the number of faces whose dimension is congruent to i modulo m with the number of all faces of P for some positive integer m and for some 0 < i < m+1. We show some classes of polytopes for which the above proportion is asymptotically equal to 1/m.