We explore star-formation histories (SFHs) of galaxies based on the evolution of the star-formation rate stellar mass relation (SFR-M*). Using data from the FourStar Galaxy Evolution Survey (ZFOURGE) in combination with far-IR imaging from the Spitzer and Herschel observatories we measure the SFR-M* relation at 0.5 < z < 4. Similar to recent works we find that the average infrared SEDs of galaxies are roughly consistent with a single infrared template across a broad range of redshifts and stellar masses, with evidence for only weak deviations. We find that the SFR-M* relation is not consistent with a single power-law of the form SFR ~ M*^a at any redshift; it has a power-law slope of a~1 at low masses, and becomes shallower above a turnover mass (M_0) that ranges from 10^9.5 - 10^10.8 Msol, with evidence that M_0 increases with redshift. We compare our measurements to results from state-of-the-art cosmological simulations, and find general agreement in the slope of the SFR-M* relation albeit with systematic offsets. We use the evolving SFR-M* sequence to generate SFHs, finding that typical SFRs of individual galaxies rise at early times and decline after reaching a peak. This peak occurs earlier for more massive galaxies. We integrate these SFHs to generate mass-growth histories and compare to the implied mass-growth from the evolution of the stellar mass function. We find that these two estimates are in broad qualitative agreement, but that there is room for improvement at a more detailed level. At early times the SFHs suggest mass-growth rates that are as much as 10x higher than inferred from the stellar mass function. However, at later times the SFHs under-predict the inferred evolution, as is expected in the case of additional growth due to mergers.
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