How-to · Updated October 2, 2026
What is the difference between a recursive and an explicit formula?
Direct answer
A recursive formula uses the previous term. An explicit formula uses n directly. For 2, 5, 8, 11 both give a 10th term of 29.
The same sequence, two formulas
| Sequence | Recursive | Explicit | 10th term |
|---|---|---|---|
| 2, 5, 8, 11 | a₁ = 2, aₙ = aₙ₋₁ + 3 | aₙ = 2 + (n - 1)(3) | 29 |
| 2, 4, 8, 16 | a₁ = 2, aₙ = aₙ₋₁ × 2 | aₙ = 2 × 2^(n-1) | 1024 |
| 1, 1, 2, 3, 5 | a₁ = 1, a₂ = 1, aₙ = aₙ₋₁ + aₙ₋₂ | Use recursive formula or Binet's formula for Fibonacci | 55 |
Geometric terms: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. Fibonacci terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
Limits
- Both formulas describe one sequence. They are not two different lists.
- The Fibonacci row needs two starting terms. The calculator’s second input is a₂ for that type.
- Binet’s formula is the closed form the calculator names for Fibonacci. The term list above is built by adding the previous two terms.
Use the calculator
The arithmetic write-up: recursive formula for an arithmetic sequence. Switch sequence type in the recursive formula calculator.