Variable Stars in Local Group Galaxies IX: Tracing the Complex History of NGC 205

We present the most extensive catalogue of variable stars in NGC 205 to date, based on archival HST/ACS imaging of a single field $\sim 4^\prime$ from the galaxy centre (GO-15902). We detect 1441 RR Lyrae stars (1064 RRab, 363 RRc, and 14 of uncertain mode) and 35 anomalous Cepheids, and measure their periods, amplitudes, and intensity-mean magnitudes in the F606W and F814W bands. The variable population is dominated by RR Lyrae stars, as expected for a predominantly old (>10 Gyr) field with no strong young component. From the red giant branch tip we derive a distance modulus of $μ= 24.68 \pm 0.06$ mag. The distribution of the RR Lyrae stars in the Bailey diagram, the fraction of RRc stars, and the presence of a substantial population of high-amplitude short-period RRab stars, indicate a range of metallicities extending to a metal-rich component, while the ratio of RR Lyrae stars to red giant branch stars indicates an extended star formation history, with only a modest fraction of the stellar mass formed in the first few Gyr. The anomalous Cepheids place NGC 205 among the slow evolvers on the relation between their frequency and host luminosity, and comparison with evolutionary models constrains their progenitors to $[\mathrm{Fe/H}] \lesssim -2.1$, indicating that metal-poor gas remained available well after the galaxy had begun to enrich. In a forthcoming paper, we will use the RR Lyrae periods and intensity-mean magnitudes to derive individual metallicities for these stars and trace the chemical enrichment history of NGC 205.