Expressions for the ULF wave growth rates are obtained by consistent study of the bounce and drift motions of geomagnetically trapped particles. Different cases depending on the relative values of bounce, drift and wave frequencies are examined. It is shown that the growth (damping) rate as a function of wave frequency has an oscillatory character with maxima associated with an averaged bounce frequency. The instability conditions are investigated. For wave frequencies exceeding the Larmor drift frequency, the instability appears if the derivative function with respect to the energy is positive over a sufficient period. If, however, the wave frequency is less than an effective Larmor drift frequency, the instability is possible even if this derivative is negative everywhere.