  We study the effective theory of slowly rotating black holes at the infinite
limit of the spacetime dimension D. This large D effective theory is obtained
by integrating the Einstein equation with respect to the radial direction. The
effective theory gives equations for non-linear dynamical deformations of the
slowly rotating black hole as effective equations. The effective equations
contain the slowly rotating Myers-Perry black hole, slowly boosted black
string, non-uniform black string and black ring as stationary solutions. We
obtain an analytic solution of the black ring by solving the effective
equations. Furthermore, by perturbation analysis of the effective equations, we
find a quasinormal mode condition of the black ring in analytic way. As a
result we confirm that thin black ring is unstable against non-axisymmetric
perturbations. We also include 1/D corrections to the effective equations and
discuss the effects by 1/D corrections.
