In this letter, a closed form approximation for the average sum rate of random beamforming is derived. It provides a near-exact approximation for arbitrary numbers of beams, users, and received SNR. The approximation is also applied to an average-sense multimode random beamforming scheme which optimizes the number of random beams without any type of instantaneous channel information. The proposed scheme shows better sum rate performance than random beamforming as well as an existing dual mode random beamforming scheme based on instantaneous channel information, while the number of feedback bits for beam index is reduced compared to random beamforming.