We conduct a theoretical study of SU(N) fermions confined by a one-dimensional harmonic potential. First, we introduce a numerical approach for solving the trapped interacting few-body problem, by which one may obtain accurate energy spectra across the full range of interaction strengths. In the strong-coupling limit, we map the SU(N) Hamiltonian to a spin-chain model. We then show that an existing, extremely accurate ansatz-derived for a Heisenberg SU(2) spin chain-is extendable to these N-component systems. Lastly, we consider balanced SU(N) Fermi gases that have an equal number of particles in each spin state for N=2,3,4. In the weak-and strong-coupling regimes, we find that the ground-state energies rapidly converge to their expected values in the thermodynamic limit with increasing atom number. This suggests that the many-body energetics of N-component fermions may be accurately inferred from the corresponding few-body systems of N distinguishable particles.