What is an explicit formula?
A formula that computes a_n from n alone.
Explicit Formula Calculator
Find the nth term of an arithmetic or geometric sequence from its explicit formula.
Choose sequence type
Enter a₁, common difference or ratio, and term index n.
About this calculator
How to use Explicit Formula Calculator, plus the formulas and logic behind the results.
Tool description
An explicit formula gives a_n directly from n without listing prior terms. Arithmetic sequences add d each step; geometric sequences multiply by r.
How to use
- Select arithmetic or geometric.
- Enter a₁ and d or r.
- Enter n ≥ 1.
- Select Calculate to view a_n.
Formulas & reference
Arithmetic: a_n = a₁ + (n − 1)dGeometric: a_n = a₁ · r^(n−1)
Examples
- a₁=3, d=5, n=10 → a₁₀ = 48
- a₁=3, r=2, n=5 → a₅ = 48
- d = 0 → constant sequence
- r = 1 → constant geometric sequence
Calculation logic
Validates n as a positive integer and applies the selected closed form.
Indexing assumes the first term is n = 1.
Explicit Formula Calculator FAQ
Common questions about this calculator and how to use the results.
What is a recursive formula?
One that defines a_n using previous term(s), e.g. a_n = a_{n−1}+d.
Can n be zero?
This tool uses n ≥ 1 for the first term.
Can r be negative?
Yes—terms will alternate sign when applicable.
How is this different from exponential growth?
Sequences are discrete in n; growth models often use continuous time.
What if I need many terms?
Compute several n values, or generate them from a₁ and d/r by hand.