  We revisit the evolution of the scale factor in a flat FRW spacetime with a
new generalized decay rule for the dynamic $\Lambda$-term under modified
theories of gravity. It analyses certain cosmological parameters and examines
their behaviours in this generalized setting which includes several decay laws
in the literature. We have also obtained observational constraints on various
model parameters and estimated the present values of cosmological parameters
$\{\Omega_{m_0}$, $\Omega_{\Lambda_0}$, $q_{0}, t_{0}$, $\omega_{0}\}$ and have
discussed with various observational results. Finite time past and future
singularities in this model are also discussed. \end{abstract}
