Last week, month etc. is the week, month just before this one. The last week, month etc. is the period of X days up to the moment of speaking.
I was in Paris last week.
I have had a cold for the last week.
I have been busy for the last three weeks.
Sometimes the last means the last in a series.
If you want to say that something happened during this period, you say that it happened in the last three weeks or during the last three weeks.
He had asked himself that question at least a thousand times in the last eight days.
Something funny happened in the last week of the holidays.
If you don't say the number, you can say
Many changes have been made in the last few years.
Or use recent instead. For example, you can say
Many changes have been made in recent years.
Interest in golf has grown rapidly in recent years.
The past+ a noun is used to refer to the period up until now. ("their activities over the past two years").
During the past two weeks ten people have died of the desease. She has been feeling tired for the past three days.
The sources are Longman Exams Dictionary, M. Swan PEU, Collins COBUILD English Usage
I think it is clear that you should use prepositions in your sentences."For"(to say how long),during,in,over (something goes on within a period of time) would do.