Well, it is surprising how someone in HR can be so irresponsible when preparing an appointment letter. Anyways, since the notice period is not mentioned in the appointment letter, you are not supposed to serve the notice period. However, if the HR is not a responsible person and has made a mistake, it does not mean you should do the same. Therefore, as a responsible employee, it is better to properly hand over the task or complete your tasks at a level where they can be handed over to another person. Make sure to complete the formalities in a positive manner before leaving the company.