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How to use
An arithmetic sequence changes by a constant amount from one term to the next. Enter the first term, common difference and term number to get the nth term, partial sum, average and an initial preview.
- Enter first term a₁.
- Enter common difference d, which may be negative or zero.
- Enter positive integer term number n.
- Review aₙ, Sₙ, the average and the first terms.
How it is calculated
Example
With a₁=5, d=3 and n=10, a₁₀=32, the first-ten-term sum is 185 and their average is 18.5.
Important notes
Term numbering starts at 1. If d=0, every term is equal. Extremely large inputs can exceed the browser’s finite-number range.
Worked examples and interpreting results
An arithmetic sequence adds the same difference each time. Starting at 2 with step 3 gives the first five terms 2, 5, 8, 11, 14.
| Case | Calculation | Result |
|---|---|---|
| Fifth term | 2 + (5 − 1) × 3 | 14 |
| Sum of first five terms | 5 × (2 + 14) ÷ 2 | 40 |
How to check the result
Term numbering starts at n = 1. Apply the common difference n − 1 times, and distinguish the last term from the total sum.
Frequently asked questions
Difference vs ratio?
Arithmetic sequences add d each step; geometric sequences multiply by r.
Can d be negative?
Yes. The sequence then decreases by a constant amount.
How is the sum found?
Sₙ=n(a₁+aₙ)/2.
Does the term index start at zero or one?
The standard formula aₙ=a₁+(n−1)d assumes the first term has n=1. If your data is indexed from zero, adjust the formula or the result will be shifted by one common difference.