What is the reason for the festivals of sukkot and pesach lasting seven days.
Is there a significance/reason to the number seven here?
"Shabbatot and yomim tovim were given just for the study of Torah" (Talmud Yerushalmi Shabbos 15:3
hence it is a kind of vacation/sabbatical for torah study.
one day would not be much of a vacation.
The Rambam in Moreh Nevukhim 3:43 says that Pesach is 7 days because a seven day period is an intermediate period between the natural day and lunar month. Just as this has significance in nature (according to Munk, this refers to the medical view at the time that 7 days was a turning point in illnesses), it is significant for the Torah, because the Torah corresponds to and perfects nature. Here is the Schwartz translation:
הוא נמשך שבעה ימים מפני שפרק הזמן של שבעה הוא פרק זמן אמצעי בין היום הטבעי וחודש הלבנה. יודע אתה שלמחזור זה יש נגיעה רבה לדברים הטבעיים. כן הוא באשר לדברים התורניים, מפני שהתורה מִדַּמָּה תמיד לטבע ומשלימה את הדברים הטבעיים באופן כלשהו, כי לטבע אין מחשבה ושיקול-דעת ואילו התורה היא קביעתו והנהגתו של האלוה הנותן את השׂכל לכל בעל-שׂכל
With regard to Sukkot, the Rambam says that it is seven days so that "so that its meaning be generally known" (Pines; Schwartz: כדי שיתפרסם הדבר).
The Rambam in the Moreh Nevuchim says:
“As that great revelation took place only on one day, so we keep its anniversary only one day; but if the eating of unleavened bread on Passover were only commanded for one day, we would not have noticed it, and its object would not have been manifest. For it frequently happens that we take the same kind of food for two or three days. But by continuing for a whole period of seven days to eat unleavened bread, its object becomes clear and evident.” -Moreh Nevuchim Chapter 43 – Page 353 in Dover Edition.
Essentially by having the Matzah for a duration of Seven days one might have a greater chance of focusing on it and considering the ideas that it is designed to teach us.