Effects of plasma nonextensivity on the nonlinear cnoidal ion-acoustic wave in unmagnetized electron-positron-ion plasma have been investigated theoretically. Plasma positrons are taken to be Maxwellian, while the nonextensivity distribution function was used to describe the plasma electrons. The known reductive perturbation method was employed to extract the KdV equation from the basic equations of the model. Sagdeev potential, as well as the cnoidal wave solution of the KdV equation, has been discussed in detail. We have shown that the ion-acoustic periodic (cnoidal) wave is formed only for values of the strength of nonextensivity (q). The q allowable range is shifted by changing the positron concentration (p) and the temperature ratio of electron to positron ( ). For all of the acceptable values of q, the cnoidal ion-acoustic wave is compressive. Results show that ion-acoustic wave is strongly influenced by the electron nonextensivity, the positron concentration, and the temperature ratio of electron to positron. In this work, we have investigated the effects of q, p, and on the characteristics of the ion-acoustic periodic (cnoidal) wave, such as the amplitude, wavelength, and frequency. KEYWORDScnoidal wave, electron-positron-ion plasma, ion-acoustic nonlinear wave, plasma nonextensivity, solitary wave INTRODUCTIONPlasma is mainly the science of universe, new accelerators, and new source of energy. Furthermore, there are many other applications of plasma that is penetrating into the technology of our ordinary life. [1,2] Plasma is a multimode medium. Depending on plasma species and condition, as well as plasma energy, a wide range of longitudinal or transverse modes in cnoidal or soliton form may generate in plasma medium. Some of them may be observed in linear regime, but there are a large number of plasma modes that will appear after taking the nonlinearity into account. Because of its high energy, the nonlinear phenomena play an essential role in plasma medium.Normal plasmas comprise of electrons, ions, and neutral particles. In the case of high-energy plasma, positrons will be produced in the medium due to pair production phenomena. The electron-positron-ion (e-p-i) plasma has an important role in the understanding of the plasmas in the early universe, [3,4] in active galactic nuclei, [5] in pulsar magnetospheres, [6,7] and in the solar atmosphere.[8] It is well known that when positrons are introduced into e-i plasma, the response of the plasma changes significantly. The presence of positrons has some application in plasma. For instance, they can be used to probe particle transport in tokamaks, and since they have sufficient lifetime, the two-component (e-i) plasma becomes three-component (e-p-i) plasma. [9][10][11][12][13] Most of the astrophysical plasmas also contain positrons in addition to electrons and ions. During the last decade, e-p-i plasma has attracted the attention of several authors. They have studied the linear and nonlinear wave propagation in e-p-i plasmas using different models. [14][1...