The phrase "a week and gradually" is not correct and lacks clarity in written English.
It may be intended to describe a process that takes a week and occurs gradually, but it needs additional context to be fully understandable.
Example: "The project will take a week and gradually improve as we implement the changes."
Alternatives: "over a week and progressively" or "in a week and slowly".
If you are sedentary, start with 10-minute bouts of activity a couple of days a week and gradually build up to longer bouts more often and at a faster pace.
After hearing of the problems, her family asked a companion, who was spending three nights a week with Ms. Hurwitz, to help her seven nights a week, and gradually added home care aides six days a week.
He found himself working 20-hour days, seven days a week, and gradually becoming less creative, less productive and less happy: "For about six months last year, for the first time really, the happiness drained away.
Take small steps; start with one home cooked meal a week and gradually increase.
Begin by exercising a set number of days a week and gradually increase the number, length, and intensity of the sessions.
Start hiking several times a week, and GRADUALLY increase the difficulty (roots, rocks, elevation gain, length) of the trails you hike.
Last season, ABC used the show four times a week, and its ratings gradually diminished.
Once he's used to this, suggest that he can also make their favorite meal one night a week and so forth, gradually extending his repertoire.
Like the 40 other subjects, she took a blue pill four times a day for a week and a half and then gradually reduced the dosage over the course of another nine days -- a total of 19 days of treatment.
After infection with S. japonicum, a defensive reaction in M. fortis commenced rapidly, increasing dramatically in the second week, and gradually decreasing three weeks after infection.
Then it increased rapidly in the following week, and gradually declined thereafter.
