The Togo international had accepted the charge for a malicious stamp on Arsenal's Dutch striker Robin van Persie during City's 4-2 Premier League home win last Saturday.
The former Arsenal player will miss Sunday's league derby away to local rivals Manchester United, a League Cup home tie against Fulham and a league match at home to West Ham United.
The FA said a separate charge of improper conduct would be considered at a later date. Adebayor's goal celebration in the same match sparked crowd trouble in which a steward was injured.
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