  Let G be a finitely generated group and (G_i) a descending chain of finite
index normal subgroups of G. Given a field K, we consider the sequence
b_1(G_i;K)/[G:G_i] of normalized first Betti numbers of G_i with coefficients
in K, which we call a K-approximation for b_1^(2)(G), the first L^2-Betti
number of G. In this paper we address the questions of when Q-approximation and
F_p-approximation have a limit, when these limits coincide, when they are
independent of the sequence (G_i) and how they are related to b_1^(2)(G). In
particular, we show that the limit of the sequence b_1(G_i;F_p)/[G:G_i] is
greater than or equal to b_1^(2)(G) under the assumptions that (G_i) has
trivial intersection and each G/G_i is a finite p-group.
