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 universes 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.
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