For noncyclic and cyclic transient diffusion and adsorption in a particle, respectively, a sequence of linear approximations is developed. In the case of noncyclic adsorption, the Fade summation on the Taylor series of the transfer function of the diffusion and adsorption system is used to derive approximations with constant coefficients. And in the case of cyclic adsorption, a frequency response matching method is developed to approximate the transfer function and applied to derive approximations with theoretical frequency-dependent coefficients. All the approximations are in the form of a state equation that consists of first-order differential equations, the number of which is the same as the approximation order. For noncyclic adsorption, the first element in the sequence, a first-order approximation, is identical to the linear driving force (LDF) equation, and for cyclic adsorption, it is identical to the approximation by Kim (1996, Chem. Engng Sci. 51, 4137). For each increment in the order, the approximation error is found to decrease roughly by an order of magnitude, and the second- or third-order approximations are shown to yield responses very close to those from the pore diffusion model. The high-order approximations for noncyclic adsorption are shown to be applicable to the cyclic adsorption over a wide range of cycle frequencies. Application of the approximations for cyclic adsorption to the cases where the effective, diffusivity varies between steps within a cycle is discussed.