  A charged, non-rotating, spherically symmetric black hole which has
cosmological constant $\Lambda$ (Reissner-Nordstr\"om+$\Lambda$ or
RN+$\Lambda$), active gravitational mass $M$ and electric charge $Q$ is studied
with exterior Friedman-Robertson-Walker (FRW) universe in (2+1) dimensional
spacetime. We find a new classes of exact solutions of the charged black hole
where the generalized Birkoff's theorem is assumed to be valid. It is found
that the cosmological constant is negative inside the black hole. We confirm it
from the geodesic equations too. The cosmological constant is found to be
dependent on $R, Q$ and $a(v)$ which correspond to the areal radius, charge, of
the black hole and the scale factor of the universe respectively. We note that
the expansion of the universe affects the size and the mass of the black hole.
An important observation is that, for an observer at infinity, both the mass
and charge of black hole increase with the contraction of the universe and
decrease with the expansion of the universe.
