Let a_(n + 1) | = | 1/2(a_n + b_n) b_(n + 1) | = | (2a_n b_n)/(a_n + b_n). Then A(a_0, b_0) = lim_(n->∞) a_n = lim_(n->∞) b_n = sqrt(a_0 b_0), which is just the geometric mean.