Explicit Formula
Functions & Advanced AlgebraAn explicit formula directly calculates any term of a sequence using its position number, without needing previous terms.
Definition
An explicit formula lets you find any term in a sequence directly, just by plugging in its position number. You do not need to know the previous terms.
Example
For the sequence $3, 7, 11, 15, \ldots$, the explicit formula is $a_n = 4n - 1$. To find the $100$th term: $a_{100} = 4(100) - 1 = 399$. No need to list all $100$ terms.
Key Insight
Explicit formulas are like shortcuts. Instead of stepping through every term to reach the 50th, you jump straight there with one calculation.
Definition
An explicit (or closed-form) formula gives $a_n$ directly as a function of $n$. Arithmetic: $a_n = a_1 + (n-1)d$. Geometric: $a_n = a_1 \cdot r^{n-1}$. These are derived from the recursive rules but computed without iteration.
Example
Arithmetic: $2, 5, 8, 11, \ldots$ $a_1 = 2$, $d = 3$. Explicit: $a_n = 2 + (n-1)(3) = 3n - 1$. Check: $a_1 = 2$, $a_4 = 11$. Find the $200$th term: $a_{200} = 3(200) - 1 = 599$.
Key Insight
Converting a recursive formula to an explicit one is a key skill. For arithmetic sequences, the explicit formula is linear in $n$. For geometric, it is exponential. This reveals the underlying function type.
Definition
An explicit formula is a closed-form expression for the general term $a_n$, defining it purely in terms of $n$ without reference to other terms. For linear recurrences, the characteristic root method yields explicit formulas involving roots of the characteristic polynomial. Not all recurrences have elementary closed forms (e.g., the partition function $p(n)$).
Example
The explicit formula for the sum of the first $n$ positive integers: $S_n = n(n+1)/2$ can be proved by induction or by Gauss's pairing argument. This is closed-form: one evaluation gives the answer for any $n$.
Key Insight
The existence of a closed-form formula is not guaranteed. Deciding whether a recursively defined sequence has a closed form is a deep question in symbolic computation, related to the theory of D-finite sequences and Liouville's theorem on integration in finite terms.