Time expressions can be tricky to use with apostrophes. Week’s, weeks’, and weeks are in fact all correct depending on the context.
Weeks is used for the plural of week. Week’s and Weeks’ are used in compound time expressions. Week’s is used with a singular time unit and weeks’ is used for a plural unit.
Week’s and Weeks’
One week’s notice
Two weeks’ notice
You need to place the apostrophe before the s when you are writing about a singular time(one day, one week, one month) and after the s when it is a plural time(two days. Three weeks, four months)
Notice that we only need to do this when we include the noun that the time expression is about. We only do this when we can replace the apostrophe with of. (Read about when to use of in English grammar here).
One week’s free insurance
Four weeks’ free insurance
She is going away in one week’s time.
She is going away in two weeks’ time.
He was docked one week’s pay.
He was docked two weeks’ pay.
You can always use of if you feel like this makes things clearer:
He was docked one week of pay.
He was docked two weeks of pay.
In this case, weeks returns to a regular singular/plural noun.
If there is no noun that the time expression is connected to then weeks acts like a regular noun, and you just need to add an s to make it plural.
I need to start work in one week.
I need to start work in two weeks.
Compound adjectives with week
Compound adjectives are when we combine two words with a hyphen to describe another noun.
He took a two-week holiday.
The package includes a one-week safari.
For compound adjectives, it is not necessary to add an s, but it is necessary to add the hyphen.