  Quantum states of a two-component Fermi trapped gas are described by
introducing an effective trap frequency, determined via variational techniques.
Closed expressions for the contribution of a contact interaction potential to
the total energy and the pairing interaction are derived. They are valid for
both few and large number of particles, given the discrete nature of the
formulation, and therefore richer than the continuous expressions, which are
perfectly matched. Pairing energies within a shell are explicitly evaluated and
its allowed values at a given energy level delimited. We show the importance of
the interaction over the trap energy as the number of particles ($N$) grows and
the temperature decreases. At zero temperature we find a polynomial dependence
of the interaction energy on the Fermi energy, whose dominant term at large $N$
corresponds with the mean field approximation result. In addition, the role of
the strength of an attractive potential on the total energy is exhibited.
