Why do you think that Arjun must continue to take antibiotics for five weeks following his release from the hospital

TLDR: Antibiotics must be taken several weeks to fully kill the bacteria. Else, surviving bacteria develops drug resistance.

Antibiotic prescription really depends on the patient's condition. Some conditions like major surgery or diseases that cause immunosuppression are more prone to bacterial infection, thus they must take antibiotics for prolonged periods of time to fight their current infections or prevent further infections.

Antibiotics, as a general rule, must be given 7 days or several weeks (depending on the bacteria/pathogen) to be sure that all the disease-causing bacteria are dead. If the drugs are taken only until symptoms fade, the surviving bacteria (now fewer in number and not causing symptoms) will develop mutations that may help resist the previously-taken antibiotic, giving rise to drug resistance.

Thus, Arjun must take the antibiotics for several weeks more (according to the doctor's orders, of course) to kill all remaining bacteria and also to prevent bacterial drug resistance. Which is really problematic, since we're slowly losing our number of effective antibiotics.