Is fasting valid on the day a brown discharge appears after ghusl following menstruation? And if the fasting was not valid, must one make up the days that followed that day?

The brownish discharge (kudra) connected to menstruation is considered part of menstruation, and ghusl (ritual bath) is not valid before ensuring purity. If a woman performs ghusl before seeing purity, her ghusl is not sufficient, and her fasting on the day she sees the kudra is not valid. She must make up the prayers she performed without ghusl. As for the fasting of the days that followed, it is valid even if she did not perform ghusl, because ghusl is not a condition for the validity of fasting. If the woman had ascertained purity before performing ghusl, then her ghusl is valid, and any kudra she sees thereafter is not considered menstruation, and her fasting and acts of worship are valid.