Terahertz (THz) quantum-cascade lasers (QCLs) are powerful radiation sources for high-resolution and high-sensitivity spectroscopy with a discrete spectrum between 2 and 5 THz as well as a continuous coverage of several GHz. However, for many applications, a radiation source with a continuous coverage of a substantially larger frequency range is required. We employed a multi-mode THz QCL operated with a fast ramped injection current, which leads to a collective tuning of equally-spaced Fabry-Pérot laser modes exceeding their separation. A continuous coverage over 72 GHz at about 4.7 THz was achieved. We demonstrate that the QCL is superior to conventional sources used in Fourier transform infrared spectroscopy in terms of the signal-to-noise ratio as well as the dynamic range by one to two orders of magnitude. Our results pave the way for versatile THz spectroscopic systems with unprecedented resolution and sensitivity across a wide frequency range.