A first-order spectral perturbation theory of resonant modes that evolve with periodically amplified optical fiber solitons is presented. In contrast with the modulational instability, these modes exhibit a linear growth in amplitude with respect to propagation and have a tuning characteristic that follows an inverse square-root dependence on the amplification period. Numerical results based on a complete solution of the nonlinear Schrödinger equation are also presented that confirm and quantify this behavior.