We study inflation in presence of Palatini $F(R)$ gravity. Since it is not always possible to find analytic solutions for the equation of motion of the auxiliary field corresponding to an arbitrary $F(R)$, we propose a new method for the computation of the inflationary observables. We test this method on $F(R) = R + \alpha R^n$ and find that a large $\alpha$ suppresses the tensor-to-scalar ratio $r$. We also notice that the configuration with $n=2$ is \emph{unstable}, while the $n>2$ case is ill-defined, with possible implications on the theoretically allowed UV behaviour of such Palatini models.