To thrive or prosper; to grow vigorously or flourish in good circumstances.
From a- (prefix meaning 'in' or 'on') plus thrive. The archaic form emphasizes the state of actively thriving, common in Middle English and Old English expressions.
Athrive shows how English once used 'a-' as a kind of intensifier prefix on lots of verbs—you could say something was 'afoot' or 'ablaze,' and athrive worked the same way, but most of these forms died out except for a few survivors.
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