Yes. You can enter values like -3/4. Enabling “Prefer exact fractions” keeps results exact when possible.

If d = 0, all terms equal a1 and the finite sum is Sn = n·a1.

How do I solve for n from a_n?

Use n = 1 + (a_n − a1)/d. If the result is not an integer, the given values do not correspond to a whole-number term index.

Arithmetic 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, a_n, or S_n.

Does each step multiply instead of add? This calculator is for arithmetic sequences, where each term adds the same difference (2, 5, 8, 11, …). If each term is multiplied by the same ratio instead, such as growing 10% every period, that's a geometric sequence. Use the Geometric Sequence & Series Calculator.

Background

An arithmetic sequence adds a constant difference d each step: a_n = a₁ + (n−1)d. An arithmetic series adds the first n terms: S_n = n/2 · (2a₁ + (n−1)d) or equivalently S_n = n/2 · (a₁ + a_n).

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: a_n = a₁ + (n−1)d
Finite sum: S_n = n/2 · (2a₁ + (n−1)d)
Difference from two terms: d = (a_k₂ − a_k₁) / (k₂−k₁)
Solve n: n = 1 + (a_n − a₁) / d
Solve a₁ from a term: a₁ = a_k − (k−1)d

Formula & Equation Used

Arithmetic term: a_n = a₁ + (n−1)d

Arithmetic sum: S_n = n/2 · (2a₁ + (n−1)d)

Solve n: n = 1 + (a_n − a₁) / d

Example Problem & Step-by-Step Solution

Example 1 — Find a_n

Let a₁=2, d=3, and n=8.

Use a_n=a₁+(n−1)d.
Compute a₈ = 2 + 7·3 = 23.

Example 2 — Find S_n

Let a₁=10, d=-2, and n=6.

Use S_n = n/2 · (2a₁ + (n−1)d).
S₆ = 6/2 · (20 + 5·(-2)) = 3 · (20 − 10) = 30

Example 3 — Solve for n (given a_n)

Let a₁=2, d=3, and a_n=23.

Use n = 1 + (a_n − 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 S_n = n·a₁.

Q: What if my sequence multiplies instead of adds?

If each term is multiplied by a constant ratio, it's a geometric sequence, not an arithmetic one. Use the Geometric Sequence & Series Calculator.