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The high number of super-Earth and Earth-like planets in the habitable zone (HZ) detected around M-dwarf stars in the last years has revealed these stellar objects to be the key for planetary radial velocity (RV) searches. Aims. Using the HARPS-N spectrograph within The HArps-n red Dwarf Exoplanet Survey (HADES) we reach the precision needed to detect small planets with a few Earth masses using the RV technique. Methods. We obtained 138 HARPS-N RV measurements between 2013 May and 2020 September of GJ 720 A, classified as an M0.5V star located at a distance of 15.56 pc. To characteriz","authors_text":"A. Bignamini, A. F. Lanza, A. Maggio, A. Petralia, A. Sozzetti, A. Su\\'arez Mascare\\~no, B. Toledo-Padr\\'on, E. Covino, E. Gonz\\'alez-\\'Alvarez, E. Poretti, G. Leto, G. Micela, G. Scandariato, I. Pagano, I. Ribas, J. I. Gonz\\'alez Hern\\'andez, J. Maldonado, L. Affer, M. Damasso, M. Perger, M. Pinamonti, P. Giacobbe, R. Rebolo, R. 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