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The Ruppeiner geometry and the fluctuation for a four-dimensional charged AdSblack hole in the framework of the extensive thermodynamics have beeninvestigated. By fixing the AdS radius of Vissers construction, the extensivethermodynamics of the RN-AdS black hole is established, and the central chargeplays the role of particle number. With the thermodynamic variables density, wereexamine the phase transition of the black hole, and the result will hold forthe black holes under different gravitational models. Moreover, we have studiedthe Ruppeiner geometry and the black hole fluctuation, and show that theincrease of central charge will reduce the fluctuation probability.

Ruppeiner geometry and the fluctuation of the RN-AdS black hole in framework of the extensive thermodynamics