All calculatorsArithmetic Sequence & Series Calculator
Compute arithmetic terms and arithmetic sums with exact fractions, optional step-by-step, a terms table, and a simple mini visual. Solve for a1, d, n, an, or Sn.
Background
An arithmetic sequence adds a constant difference d each step: an = a1 + (n−1)d. An arithmetic series adds the first n terms: Sn = n/2 · (2a1 + (n−1)d) or equivalently Sn = n/2 · (a1 + an).
How to use this calculator
- Choose what you want to compute (term, sum, solve n, or solve from two terms).
- Enter the required values.
- Click Calculate to get the answer plus (optional) steps, table, and mini visual.
- Optional: keep Prefer exact fractions on to avoid rounding issues.
How this calculator works
- Term formula: an = a1 + (n−1)d
- Finite sum: Sn = n/2 · (2a1 + (n−1)d)
- Difference from two terms: d = (ak₂ − ak₁) / (k₂−k₁)
- Solve n: n = 1 + (an − a1) / d
- Solve a1 from a term: a1 = ak − (k−1)d
Formula & Equation Used
Arithmetic term: an = a1 + (n−1)d
Arithmetic sum: Sn = n/2 · (2a1 + (n−1)d)
Solve n: n = 1 + (an − a1) / d
Example Problem & Step-by-Step Solution
Example 1 — Find an
Let a1=2, d=3, and n=8.
- Use an=a1+(n−1)d.
- Compute a8 = 2 + 7·3 = 23.
Example 2 — Find Sn
Let a1=10, d=-2, and n=6.
- Use Sn = n/2 · (2a1 + (n−1)d).
- S6 = 6/2 · (20 + 5·(-2)) = 3 · (20 − 10) = 30
Example 3 — Solve for n (given an)
Let a1=2, d=3, and an=23.
- Use n = 1 + (an − a1) / d.
- n = 1 + (23 − 2)/3 = 1 + 21/3 = 8
Frequently Asked Questions
Q: What makes a sequence arithmetic?
Each term increases or decreases by the same constant difference d.
Q: Can I use fractions?
Yes. Turn on Prefer exact fractions to keep results exact when possible.
Q: What if d = 0?
Then every term equals a1, and Sn = n·a1.
