Let k be a number field and let p ≥ 2 be prime. We call K/k a cyclic p-tower if Gal(K/k) ≃ Z/p N Z, N ≥ 2. We give an elementary proof of a stability theorem for generalized p-class groups Xn of the layers kn (Theorem 3.1). Using generalizations of Chevalley's formula (Gras, J. Math. Soc. Japan 46(3) (1994), Proc. Math. Sci. 127(1) (2017)), we improve results by Fukuda (1994), Li-Ouyang-Xu-Zhang (2020), Mizusawa-Yamamoto (2020) and others, whose techniques are based on Iwasawa's theory, Galois theory of pro-p-groups or specific methods. The main difference, compared to these works, is that we introduce an explicit parameter λ ≥ 0 giving formulas of the form #X n = #X 0 • p λ•n , for all n ∈ [0, N ], as soon as this equality holds for n = 1.