Recursive formulas have two parts: the first element and a formula relating the value of one element to the next. For example, in the sequence "1, 3, 5, 7, 9, ..." the first element is 1; the formula is the current element equals the last element plus two.
The interesting thing about recursive formulas is that they are self-referential. The value of a term in the sequence depends on the value before it, and that value depends on the value before it, and so on — back to the first element. To determine the value of any term, determine the value of all terms before it, back to the first term, which is fixed. This makes them difficult to calculate as the series grows longer. In general, however, equivalent explicit functions exist for calculating members of the series without going all the way back to the beginning.