Final answer:
The most appropriate management for a lodged file fragment in the apical third of the canal during endodontic treatment is to resect the apical section of the root containing the separated instrument.
Step-by-step explanation:
In this scenario, the most appropriate management would be to resect the apical section of the root containing the separated instrument. This involves removing the section of the root canal that contains the lodged file fragment. By doing so, the canal can be properly cleaned and shaped, and the treatment can proceed without any complications.
Extracting the tooth or performing an apicoectomy and placing a retrograde filling may be more invasive procedures that are not necessary in this case since no radiographic changes at the apex are evident. Completing the root canal filling and monitoring at recall examination may also not be ideal as leaving the separated instrument in the apical third of the canal can lead to persistent infection and possible failure of the treatment.