Multi-layer structures possess highly geometrically tunable optical resonances to change the surface plasmon frequencies in the wide range. A simple analytic model is presented to describe the plasmon hybridization of multi-layer metamaterial structures with multipole resonances. We theoretically demonstrate that the number of plasmon modes increases with the number of layers, which results in the enlargement of the localized resonant optical band gap. To explore the interaction between incident fields and multi-layer structures, we derive the closed-form perturbed field in terms of a generalized polarization matrix, whose dimension matches the number of layers. This enables us to obtain the eigenmode frequencies and their corresponding eigenfunctions for three-dimensional multi-layer concentric spheres and two-dimensional multi-layer concentric cylinders in a vacuum setting. These findings offer nanoscientists a general design principle that can be applied to facilitate the proper selection of the metamaterial parameters, thereby inducing the desired resonance and enabling customized applications to be realized. •Theoretical breakthrough: An algebraic framework for analyzing the plasmon modes is derived for multi-layered structures.•Numerically, it is quite easy to compute all the eigen modes by finding the roots to the exact polynomial derived.•Numerical findings: increase of layers could help to enlarge localized resonant band gap.