  Considering a three-dimensional $C-$metric, we obtain the exact accelerating
black holes in the $F(R)$ theory of gravity coupled with and without a matter
field. First, we extract uncharged accelerating AdS black hole solutions in
$F(R)$ gravity. Then, we study the effects of various parameters on metric
function, roots, and the temperature of these black holes. The temperature is
always positive for the radii less than $\frac{1}{\alpha }$, and it is negative
for the radii more than $\frac{1}{\alpha }$. We extend our study by coupling
nonlinear electrodynamics as a matter filed to $F(R)$ gravity to obtain charged
black holes in this theory. Next, we evaluate the effects of different
parameters such as the electrical charge, accelerating parameter, angular,
$F(R)$ gravity, and scalar curvature on the obtained solutions, roots, and
temperature of three-dimensional charged accelerating AdS black holes. The
results indicate that there is a root in which it depends on various
parameters. The temperature of these black holes is positive after this root.
