  We present a new theoretical framework for describing an impurity in a
trapped Bose system in one spatial dimension. The theory handles any external
confinement, arbitrary mass ratios, and a weak interaction may be included
between the Bose particles. To demonstrate our technique, we calculate the
ground state energy and properties of a sample system with eight bosons and
find an excellent agreement with numerically exact results. Our theory can thus
provide definite predictions for experiments in cold atomic gases.
