How-to · Updated October 2, 2026
How do you write a recursive formula for an arithmetic sequence?
Direct answer
For 2, 5, 8, 11, … the recursive formula is a₁ = 2, aₙ = aₙ₋₁ + 3. The 10th term is 29.
The first ten terms
Each term is the one before it plus 3. The explicit formula for the same list is aₙ = 2 + (n - 1)(3).
| n | Term |
|---|---|
| 1 | 2 |
| 2 | 5 |
| 3 | 8 |
| 4 | 11 |
| 5 | 14 |
| 6 | 17 |
| 7 | 20 |
| 8 | 23 |
| 9 | 26 |
| 10 | 29 |
A second sequence
3, 7, 11, 15 uses a difference of 4. The formula is a₁ = 3, aₙ = aₙ₋₁ + 4. The 5th term is 19. The terms through n = 5 are 3, 7, 11, 15, 19.
Limits
- This rule is for a constant difference. A constant multiplier is a geometric sequence.
- The recursive form needs the previous term. The explicit form jumps to term n.
- A Fibonacci-style rule adds the previous two terms and needs two starting values.
Use the calculator
How the two formulas differ: recursive vs explicit formula. Change the first term and the difference in the recursive formula calculator.