  The $f(R)$ theory of gravitation developed perturbatively around the general
theory of relativity with cosmological constant (the \text{$\Lambda$}CDM model)
in a flat FLWR geometry is considered. As a result, a general explicit
cosmological solution that can be used for any model with an arbitrary, but
well-defined, $f(R)$ function (just satisfying given perturbation conditions)
is derived. This perturbative solution shows how the Hubble parameter $H (t)$
depends on time (along with the cosmological constant and the matter density)
to adapt to the evolution of the Universe. To illustrate, this approach is
applied to some specific test models. One of these models appears to be more
realistic as it could describe three phases of the Universe's evolution.
  Despite the fact that the perturbation is applied for a flat FLWR geometry
(according to the current cosmological observation) indicates that the obtained
solution can mainly describe the evolution of the late Universe, it may also
work for an early Universe. As a next step, the present method can be applied
to the case with a more general FLRW geometry to increase the precision of the
description of different stages in the evolution of the Universe.
  Finally, it is shown that in a desription of the Universe's evolution the
perturbative $f(R)$-theory can be considered as an effective GR with the
cosmological constant $\Lambda$ replaced by an effective parameter $
\Lambda_{eff}[\rho(t)] $. This trick leads to a simpler way of solving an
$f(R)$-theory regardless its specific form.
