Recursive Formulas

Algebra 2 · Axiom Academy

Define each term from the one before it — a starting value plus a rule like a_n = a_ n-1 + d . 1. A Term From the One Before It A recursive definition has exactly two parts: a seed that starts the sequence, and a rule that produces each new term from the previous one. Take the seed a_1 = 3 and the rule "add 2 to the term before you": Initial condition — the seed the chain starts from Recursive rule — each term from the previous one The recursive skeleton — previous term rule next term — is the same for the two most common sequences. Only the rule changes: an arithmetic sequence adds a fixed common difference d , while a geometric sequence multiplies by a fixed common ratio r . Seed a_1 = 3 , difference d = 2 : gives 3, 5, 7, 9, 11 . The gap between terms is constant, so the values grow in a straight line. Seed a_1 = 2 , ratio r = 3 : gives 2, 6, 18, 54, 162 . Each term is a fixed multiple of the last, so the values explode upward. 3. Reaching Back Two: Fibonacci A recursive rule can depend on more than one previous term. The famous Fibonacci sequence adds the two terms before each new one — so it needs two seeds to get started. Two seeds, then each term is the sum of the previous two. Each term looks back two steps, so you build from the seeds outward: Because a two-back rule reaches to F_ n-1 and F_ n-2 , a single seed is not enough — you must supply both F_1 and F_2 .

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