Recursive and Explicit, Both Ways

The big idea

There are two ways to describe a list. A recursive rule is the stairs: it gives the starting term and how to get each next term from the one before, Next equals Now plus 12 starting at 4, or in formal notation a sub n plus 1 equals a sub n plus 12 with a sub 1 equals 4. An explicit rule is the elevator: it finds any term straight from its term number. To go from recursive to explicit, read the starting term, decide whether the rule adds (arithmetic, d) or multiplies (geometric, r), and write a sub 1 plus d times n minus 1, or a sub 1 times r to the n minus 1. To go from explicit to recursive, put in n = 1 for the start, read d or r, and write the start with the next-from-now rule. Recursive is easy to read and good for the next few terms; explicit finds the 100th term in one step. Some lists, like the Fibonacci numbers, have a simple recursive rule and no easy explicit one, which is why you learn both.