Multiple-Input Multiple-Output (MIMO) radar system allows each antenna element to transmit a different waveform. This waveform diversity can be utilized to enhance the beampattern design, in particular effective management of radar radiation power in directions of interest. In this paper, we address the problem of designing a beampattern for Multiple-Input Multiple-Output (MIMO) radar, which in turn is determined by the transmit waveform. While unconstrained design is straightforward, a key open challenge is enforcing the constant modulus constraint on the radar waveform. Existing beampattern design methods that address constant modulus invariably lead to a stiff trade-off between analytical tractability (achieved by relaxations and approximations) and realistic design that exactly achieves constant modulus but is computationally burdensome. A new approach is proposed in our work, which involves solving the hard non-convex problem of beampattern design using a sequence of convex Equality Constrained Quadratic Programs, each of which has a closed form solution. Constant modulus is achieved at convergence, which we prove formally is possible under mild and realistic assumptions. We evaluate the proposed successive closed forms (SCF) algorithm against state of the art MIMO beampattern design techniques and show that SCF breaks the trade-off between desirable performance and the associated computation cost.