late 14c., in church music, a composition said or sung after the Alleluia and before the Gospel, from Old French sequence "answering verses" (13c.) and directly from Medieval Latin sequentia "a following, a succession," from Latin sequentem (nominative sequens), present participle of sequi "to follow" (from PIE root *sekw- (1) "to follow").
In Church use, a partial loan-translation of Greek akolouthia, from akolouthos "following." By 1570s in the general sense of "a series of things following in a certain order, a succession," also in cards, "a run of three or more consecutive numbers of the same suit." By 1580s as "order of succession." In biochemistry in reference to nucleic acids, by 1959.