You can only do 1 triumph per week. If someone in your fireteam is on a different week it may screw it up. So if you finished week 1 triumph this week you have to wait for the next weekly reset to do week 2, ect.

There were 5 from week1, 6 in week2, and more in week3. Once you find all 5 from week1, then week2’s will become active following the weekly reset, then week3 after you find all from week2.