"The population increased a lot in the last century."
Translation:Geçen asırda nüfus çok arttı.
I think it is very similar to English from this aspect. You can say last year in English as well as you can say 'geçen sene' in Turkish.
I would say -de/-da is half-optional with time adverbs. So I mean the more common the expression the less common -de/-da. Also the formal/scientific/business/serious language usually requires you to use it. So geçen ayda (in the last month) sounds more formal than geçen ay (last month)