A treatment of linearized gravity with the Einstein-Gauss-Bonnet interaction terms in D≥ 5 is given in the Randall-Sundrum brane background. This Letter has outlined some interesting features of the brane world gravity and Newtonian potential correction with the GB interaction term. We find that the GB coupling α renormalizes the effective four-dimensional Newton constant on the brane, and also additionally contributes to the correction term of the Newtonian potential. Indeed, the GB term does

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... ot affect the massless graviton mode and the Einstein gravity on the brane, and quite interestingly, such term in D≥ 5 appears to give more information about the necessary boundary condition(s) to be satisfied by the zero mode wavefunction on the brane(s). small (and preferably positive) GB coupling α.