  The objective of the present paper is to study 4-dimensional weakly Ricci
symmetric spacetimes $(WRS)_4$ with non-zero constant Ricci scalar. We prove
that such a $(WRS)_4$ satisfying $F(R)$-gravity field equations represents a
perfect fluid with vanishing vorticity. Some energy conditions are studied
under the current setting to constrain the functional form of $F(R)$. We
examine a couple of popular toy models in $F(R)$-gravity, $F(R)=e^{\alpha R}$
where $\alpha$ is constant and $F(R)=R-\beta \tanh(R)$, $\beta$ is a constant.
We also find that the equation of state parameter (EoS) in both models supports
the universe's accelerating behavior, i.e., $\omega=-1$. According to the
recently suggested observations of accelerated expansion, both cases define
that the null, weak, and dominant energy conditions justify their requirements
while the strong energy conditions violate them.
