Writing a Recursive Rule for a Sequence (F 603)

A recursive rule describes a sequence the way you'd describe a habit: whatever you had last time, do this to it. Instead of a formula that jumps straight to term 50, you get a relationship between consecutive terms, written a(n) = (something)·a(n-1) + (something). This beats a closed formula whenever the pattern is built by repeated action rather than by a clean function of n — it mirrors how the sequence actually grows, term by term, which makes it easier to spot and easier to check.

To find the rule, compare consecutive terms. If the differences are constant, you're adding. If the ratios are constant, you're multiplying. If neither alone works, test both together: multiply by some factor, then add or subtract a constant, and check that the same two operations carry every term to the next. A rule isn't finished once you have that step, though. a(n) = a(n-1) + 5 could describe 3, 8, 13, 18 or 100, 105, 110, 115 — the step alone doesn't fix the list. You must also state the first term, a(1), to pin down one specific sequence.

This topic sits right next to explicit formulas and sums: once you have a recursive rule, you can sometimes convert it to an explicit formula for arithmetic or geometric sequences, or use it directly to compute sums term by term. Recursive thinking also connects forward to working backward — if you know a later term and the rule, you can undo the operations in reverse order to recover earlier terms, including a(1).