Explicit Formula
An explicit formula directly calculates any term of a sequence using its position number, without needing previous terms.
Definition
An explicit (or closed-form) formula lets you find any term of a sequence directly by plugging in its position number, with no need to know the previous terms: for arithmetic sequences, $a_n = a_1 + (n-1)d$, and for geometric sequences, $a_n = a_1 \cdot r^{n-1}$, both derived from the recursive rules but computed without iteration. More generally, an explicit formula defines the general term $a_n$ purely in terms of $n$; for linear recurrences, the characteristic root method yields explicit formulas involving roots of the characteristic polynomial, but not all recurrences have elementary closed forms, such as the partition function $p(n)$.
Example
For the sequence $3, 7, 11, 15, \ldots$, the explicit formula is $a_n = 4n - 1$, so the $100$th term is $a_{100} = 4(100) - 1 = 399$ with no need to list all $100$ terms. For $2, 5, 8, 11, \ldots$ with $a_1 = 2$, $d = 3$: the explicit formula is $a_n = 3n - 1$, giving a $200$th term of $599$. The explicit formula for the sum of the first $n$ positive integers, $S_n = n(n+1)/2$, can be proved by induction or Gauss's pairing argument, evaluating instantly for any $n$.
Key Insight
Explicit formulas are like shortcuts: instead of stepping through every term to reach the $50$th, you jump straight there with one calculation, and converting a recursive formula to an explicit one is a key skill, revealing that arithmetic sequences are linear in $n$ while geometric sequences are exponential. The existence of a closed-form formula is not guaranteed: deciding whether a recursively defined sequence has one is a deep question in symbolic computation, related to the theory of D-finite sequences and Liouville's theorem on integration in finite terms.