  By means of the density matrix renormalization group (DMRG) method, the
magnetic properties of the J-J-J$^{\prime}$ quantum Heisenberg chains with spin
$S=1/2$, 1, 3/2 and 2 in the ground states are investigated in the presence of
a magnetic field. Two different cases are considered: (a) when $J$ is
antiferromagnetic and $J^{\prime}$ is ferromagnetic (i.e. the AF-AF-F chain),
the system is a ferrimagnet. The plateaus of the magnetization are observed. It
is found that the width of the plateaus decreases with increasing the
ferromagnetic coupling, and disappears when $% J^{\prime}/J$ passes over a
critical value. The saturated field is observed to be independent of the
ferromagnetic coupling; (b) when $J$ is ferromagnetic and $J^{\prime}$ is
antiferromagnetic (i.e. the F-F-AF chain), the system becomes an
antiferromagnet. The plateaus of the magnetization are also seen. The width of
the plateaus decreases with decreasing the antiferromagnetic coupling, and
disappears when $J^{\prime}/J $ passes over a critical value. Though the ground
state properties are quite different, the magnetization plateaus in both cases
tend to disappear when the ferromagnetic coupling becomes more dominant.
Besides, no fundamental difference between the systems with spin half-integer
and integer has been found.
