Extended gravity theories and the Einstein--Hilbert action

I discuss the relation between arbitrarily high-order theories of gravity and scalar-tensor gravity at the level of the field equations and the action. I show that (2n+4) -order gravity is dynamically equivalent to Brans-Dicke gravity with an interaction potential for the Brans-Dicke field and n further scalar fields. This scalar-tensor action is then conformally equivalent to the Einstein-Hilbert action with n+1 scalar fields. This clarifies the nature and extent of the conformal equivalence between extended gravity theories and general relativity with many scalar fields.