Recursive Formula
A recursive formula defines each term of a sequence using one or more previous terms, along with a starting value.
Definition
A recursive formula tells you how to find the next term of a sequence using the term (or terms) before it, always starting from a given first term and building forward. Arithmetic sequences use $a_n = a_{n-1} + d$, geometric sequences use $a_n = r \cdot a_{n-1}$, and the Fibonacci sequence uses two previous terms, $a_n = a_{n-1} + a_{n-2}$ with $a_1 = 1$, $a_2 = 1$. Formally, a recursive formula defines a sequence via a recurrence relation $a_n = F(a_{n-1}, \ldots, a_{n-k})$ with $k$ initial conditions; linear recurrences with constant coefficients can be solved in closed form using the characteristic polynomial, and if $r_1, \ldots, r_k$ are its distinct roots, the general solution is $a_n = \sum C_i r_i^n$.
Example
With $a_1 = 3$ and $a_{n+1} = a_n + 5$: $3$, then $3+5=8$, then $8+5=13$, then $13+5=18$, generating $3, 8, 13, 18, \ldots$ A geometric recursive formula $a_1 = 5$, $a_n = 3 \cdot a_{n-1}$ gives $5, 15, 45, 135, 405, \ldots$, though finding $a_{10}$ this way requires computing every previous term. For Fibonacci, $a_n = a_{n-1} + a_{n-2}$, the characteristic equation $x^2 = x + 1$ has roots $\phi = (1 + \sqrt{5})/2$ and $\psi = (1 - \sqrt{5})/2$, giving Binet's formula $a_n = (\phi^n - \psi^n) / \sqrt{5}$.
Key Insight
Recursive means "defined in terms of itself": you need a starting point and a rule to step forward, without the first term you cannot begin, and recursive formulas are often simpler to write but require computing every previous term to reach a specific one, unlike explicit formulas which are more efficient for large $n$. The connection between linear recurrences and characteristic polynomials mirrors the connection between linear ODEs and their characteristic equations, revealing a deep structural parallel between discrete and continuous mathematics.