  We report semiconductor to metal-like crossover in temperature dependence of
resistivity ($\rho$) due to the switching of charge transport from bulk to
surface channel in three-dimensional topological insulator
Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$. Unlike earlier studies, a much
sharper drop in $\rho$($T$) is observed below the crossover temperature due to
the dominant surface conduction. Remarkably, the resistivity of the conducting
surface channel follows a rarely observable $T^2$ dependence at low temperature
as predicted theoretically for a two-dimensional Fermi liquid system. The field
dependence of magnetization shows a cusp-like paramagnetic peak in the
susceptibility ($\chi$) at zero field over the diamagnetic background. The peak
is found to be robust against temperature and decays linearly with field from
its zero-field value. This unique behavior of $\chi$ is associated with the
spin-momentum locked topological surface state of
Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$. The reconstruction of surface state
with time is clearly reflected through the reduction of peak height with the
age of the sample.
