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We report the synthesis and systematic studies of a new layered ternary telluride TaPdTe5 with quasi-one-dimensional PdTe2 chains. This compound crystalizes in a layered orthorhombic structure with space group Cmcm. Analysis of its curved field-dependent Hall resistivity, using the two-band model, indicates the hole-dominated transport with a high mobility ${\mu}_h$ = 2.38 $\times$ 10$^3$ cm$^2$ V$^{-1}$ s$^{-1}$ at low temperatures. The in-plane magnetoresistance (MR) displays significant anisotropy with field applied along the crystallographic $b$ axis. The MR with the current applied along the $c$-axis is also measured in high magnetic fields up to 51.7 T. Remarkably, it follows a power-law dependence and reaches (9.5 $\times$ 10$^3$)% at 2.1 K without any signature of saturation. The De Haas-van Alphen oscillations show a small Fermi-surface pocket with a nontrivial Berry phase. The Shubnikov-de Haas (SdH) oscillations are detected at low temperatures and under magnetic fields above 28.5 T. Two effective masses $m^*$ (0.26$m_e$ and 0.41$m_e$) are extracted from the oscillatory SdH data. Our first-principles calculations unveil a topological Dirac cone in its surface states, and, in particular, the topological index indicates that TaPdTe$_5$ is a topologically nontrivial material., Comment: 12 pages,6 figures