The initial value is the fine owed when the number of overdue days is zero. Based on the pattern in the table, each day overdue increases the fine by 15 cents (the difference in fines from one day to the next).
If we extend the pattern for 0 days:
- For 2 days, the fine is 30 cents.
- For 1 day, the fine would be 30 cents - 15 cents = 15 cents.
- For 0 days, the fine would be 15 cents - 15 cents = 0 cents.
So, the initial value, or fine owed for an overdue library book after 0 days, is 0 cents.