  In this paper by deriving the Modified Friedmann equation in the Palatini
formulation of $R^2$ gravity, first we discuss the problem of whether in
Palatini formulation an additional $R^2$ term in Einstein's General Relativity
action can drive an inflation. We show that the Palatini formulation of $R^2$
gravity cannot lead to the gravity-driven inflation as in the metric formalism.
If considering no zero radiation and matter energy densities, we obtain that
only under rather restrictive assumption about the radiation and matter energy
densities there will be a mild power-law inflation $a(t)\sim t^2$, which is
obviously different from the original vacuum energy-like driven inflation. Then
we demonstrate that in the Palatini formulation of a more generally modified
gravity, i.e., the $1/R+R^2$ model that intends to explain both the current
cosmic acceleration and early time inflation, accelerating cosmic expansion
achieved at late Universe evolution times under the model parameters satisfying
$\alpha\ll\beta$.
