Could you tell me about the right preposition in this phrase: on/in even weeks in contrast with on/in odd weeks? Besides, is the determiner THE necessary before the noun WEEK and why is it or is it not?
The preposition you want to use is
on even weeks
means on the second, forth, sixth, eighth, etc week of the year.
In your usage "on" = "during", since during those weeks you do something, or "at that appointed time".
There is bus service on even weeks.
there is bus service every other week.
The clock chimes on the hour.
the clock chimes at the hour.