  We derive the exact Helmholtz free energy (HFE) of the standard and staggered
one-dimensional Blume-Emery-Griffiths (BEG) model in the presence of an
external longitudinal magnetic field. We discuss in detail the thermodynamic
behavior of the ferromagnetic version of the model, which exhibits magnetic
field-dependent plateaux in the $z$-component of its magnetization at low
temperatures. We also study the behavior of its specific heat and entropy, both
per site, at finite temperature. The degeneracy of the ground state, at $T=0$,
along the lines that separate distinct phases in the phase diagram of the
ferromagnetic BEG model is calculated, extending the study of the phase diagram
of the spin-1 antiferromagnetic (AF) Ising model in[S.M. de Souza and M.T.
Thomaz, J. of Mag. and Mag. Mat. {354} (2014) 205]. We explore the implications
of the equality of phase diagrams, at $T=0$, of the ferromagnetic BEG model
with $\frac{K}{|J|} = -2$ and of the spin-1 AF Ising model for $\frac{D}{|J|} >
\frac{1}{2}$.
