Sometimes we can define a sequence recursively. That is, we can describe how to calculate the next term in a sequence if we know the previous term.

Here’s a sequence: 6, 10, 14, 18, 22, . . . This is an arithmetic sequence, where each term is 4 more than the previous term. Since sequences are functions, let's call this sequence , and then we can use function notation to write . Here, is the term, is the previous term, and + 4 represents the rate of change since is an arithmetic sequence.

When we define a function recursively, we also must say what the first term is. Without that, there would be no way of knowing if the sequence defined by started with 6 or 81 or some other number. Here, one possible starting term is . It is possible to start sequences with input values other than 1, and the starting value often depends on what the sequence represents.

Combining this information gives the recursive definition for : and for , where is an integer. We include the at the end since the value of at 1 is already given and the other terms in the sequence are generated by inputting integers larger than 1 into the definition.