We address the question of how fast the available rational torsion on abelian varieties over ℚ increases with dimension. The emphasis will be on the derivation of sequences of torsion divisors on hyperelliptic curves. Work of Hellegouarch and Lozach (and Klein) may be made explicit to provide sequences of curves with rational torsion divisors of orders increasing linearly with respect to genus. The main results in §2) are applications of a new technique which provide sequences of hyperellipti...