Most existing treatments of induced dipoles in polarizable molecular mechanics force field calculations use either the self-consistent variational method, which is solved iteratively, or the "direct" approximation that is non-iterative as a result of neglecting coupling between induced dipoles. The variational method is usually implemented using assumptions that are only strictly valid under tight convergence of the induced dipoles, which can be computationally demanding to enforce. In this work, we discuss the nature of the errors that result from insufficient convergence and suggest a strategy that avoids such problems. Using perturbation theory to reintroduce the mutual coupling into the direct algorithm, we present a computationally efficient method that combines the precision of the direct approach with the accuracy of the variational approach. By analyzing the convergence of this perturbation series, we derive a simple extrapolation formula that delivers a very accurate approximation to the infinite order solution at the cost of only a few iterations. We refer to the new method as extrapolated perturbation theory. Finally, we draw connections to our previously published permanent multipole algorithm to develop an efficient implementation of the electric field and Thole terms and also derive some necessary, but not sufficient, criteria that force field parameters must obey.