I’ve got an idea…let’s see if I can work it out.
Day one: cut off a segment and pay them. They now have one segment.
Day two: cut off a piece 2 segments big and give them that and ask for the first days gold back – they now have two segments
Day three: Give them the single segment again. They now have three segments
Day four: give them the remaining piece with 4 segments and ask for all the other bits back. They now have 4 segments.
Day five: give them the single segment. They now have 5 segments.
Day 6: Give them the ‘2 segment’ piece and ask for the single segment back. They now have 6 segments.
Day 7: give them the single segment. They now have 7 segments.
I’ve done it!
Sorry for solving your riddle so quickly. I think you need to have a Eureka moment and it came to me immediately. Usually I’m rubbish at puzzles!