It is generally believed that eruptive phenomena in the solar atmosphere such as solar flares and coronal mass ejections (CMEs) occur in the solar active regions with complex magnetic structures. Magnetic helicity has been recognized as a useful parameter to measure the complexity such as twists, kinks, and inter-linkages of magnetic field lines. The objective of this study is to understand a long-term (a few days) variation of magnetic helicity in active regions and its relationship with the energy buildup and instability leading to flares and CMEs. Statistical studies of flare productivity and magnetic helicity injection in about 400 active regions were carried out. The temporal variation of magnetic helicity injected through the photosphere of active regions was also examined related to 46 CMEs. The main findings in this study are as follows: (1) the study of magnetic helicity for active regions producing major flares and CMEs indicates that there is always a significant helicity injection through the active-region photosphere over a long period of 0.5 - a few days before the flares and CMEs; (2) for the 30 CMEs under investigation, it is found that there is a fairly good correlation (linear correlation coefficient of 0.71) between the average helicity injection in the CME-productive active regions and the CME speed. Beside the scientific contribution, a major impact of this study is the observational discovery of a characteristic variation pattern of magnetic helicity injection in flare/CME-productive active regions which can be used for the improvement of solar eruption forecasting.