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We show, in detail, that the only non-trivial black hole (BH) solutions for aneutral as well as a charged spherically symmetric space-times, using the class${\textit F(R)}={\textit R}\pm{\textit F_1 (R)} $, must-have metric potentialsin the form $h(r)=\frac{1}{2}-\frac{2M}{r}$ and$h(r)=\frac{1}{2}-\frac{2M}{r}+\frac{q^2}{r^2}$. These BHs have a non-trivialform of Ricci scalar, i.e., $R=\frac{1}{r^2}$ and the form of ${\textit F_1(R)}=\mp\frac{\sqrt{\textit R}} {3M} $. We repeat the same procedure for(Anti-)de Sitter, (A)dS, space-time and got the metric potentials of neutral aswell as charged in the form $h(r)=\frac{1}{2}-\frac{2M}{r}-\frac{2\Lambda r^2}{3} $ and $h(r)=\frac{1}{2}-\frac{2M}{r}+\frac{q^2}{r^2}-\frac{2\Lambda r^2}{3} $, respectively. The Ricci scalar of the (A)dS space-times has the form${\textit R}=\frac{1+8r^2\Lambda}{r^2}$ and the form of ${\textitF_1(R)}=\mp\frac{\textit 2\sqrt{R-8\Lambda}}{3M}$. We calculate thethermodynamical quantities, Hawking temperature, entropy, quasi-local energy,and Gibbs-free energy for all the derived BHs, that behaves asymptotically asflat and (A)dS, and show that they give acceptable physical thermodynamicalquantities consistent with the literature. Finally, we prove the validity ofthe first law of thermodynamics for those BHs.

Uniqueness of non-trivial spherically symmetric black hole solution in special classes of F(R) gravitational theory