recursive formula
A recursive formula builds a sequence step by step, defining each new term using the terms that came before it — like a recipe that says “to get tomorrow's value, take today's and add 3.” You climb the ladder one rung at a time.
Such a definition has two parts: one or more starting values (the base case, e.g. a_1 = 2) and a recurrence rule that produces the next term from earlier ones (e.g. a_n = a_(n−1) + 3). Without a base case the rule has nothing to start from.
Recursion is natural and easy to write, but it has a cost: to find the 100th term you must compute all 99 before it. An explicit formula, when one exists, lets you jump straight to any term. The famous Fibonacci sequence, a_n = a_(n−1) + a_(n−2), is defined this way.
a_1 = 2, a_n = a_(n−1) + 3 gives 2, 5, 8, 11, 14, ... — the same arithmetic sequence an explicit formula would describe as a_n = 3n − 1.
Base case plus recurrence rule, generating the sequence one term at a time.