  A Heisenberg spin-$s$ chain with alternating ferromagnetic ($-J_1^F<0$) and
antiferromagnetic ($J_1^A>0$) nearest-neighbor (NN) interactions, exhibits the
Dimer and spin-$2s$ Haldane phases in the limits $J_1^F/J_1^A \rightarrow 0$
and $J_1^F/J_1^A \rightarrow \infty$ respectively. These two phases are
understood to be topologically equivalent. Induction of the frustration through
the next nearest-neighbor ferromagnetic interaction ($-J_2^F<0$) produces a
very rich quantum phase diagram. With frustration, the whole phase diagram is
divided into a ferromagnetic (FM) and a nonmagnetic (NM) phase. For $s=1/2$,
the full NM phase is seen to be of Haldane-Dimer type, but for $s>1/2$, a
spiral phase comes between the FM and the Haldane-Dimer phases. The study of a
suitably defined string-order parameter and spin-gap at the phase boundary
indicates that the Haldane-Dimer and spiral phases have different topological
characters. We also find that, along the $J_2^F=\frac 12 J_1^F$ line in the NM
phase, an NN dimer state is the {\it exact} groundstate, provided
$J_1^A>J_C=\kappa J_1^F$ where $\kappa \le s + h$ for applied magnetic field
$h$. Without magnetic field, the position of $J_C$ is on the FM-NM phase
boundary when $s=1/2$, but for $s>1/2$, the location of $J_C$ is on the phase
separation line between the Haldane-Dimer and spiral phases.
