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Arithmetic Sequence Calculator

Calculate an arithmetic sequence online — free and step by step. Find the nth term aₙ = a₁ + (n−1)·d and the sum Sₙ with the full working.

Quick answer
How do you calculate an arithmetic sequence?
The nth term is aₙ = a₁ + (n−1)·d, with first term a₁ and constant difference d. Example: a₁ = 3, d = 5, n = 10 → 3 + 9·5 = 48. The sum of the first n terms is Sₙ = n/2 · (2a₁ + (n−1)d) — for the same example 10/2 · (6 + 45) = 255.
The tool

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What is the value of the nth term?
Comma or dot as decimal separator, negative values allowed.
Step-by-step
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HowTo

Calculate an arithmetic sequence — step by step

Using a₁ = 3, d = 5 — nth term and sum
  1. 1
    Step 1 of 4

    Identify the first term and difference

    Read off a₁ (the first term) and d (the constant difference between consecutive terms). Here a₁ = 3 and d = 5, so 3, 8, 13, 18, …

  2. 2
    Step 2 of 4

    Apply the nth-term formula

    aₙ = a₁ + (n−1)·d. For the 10th term: 3 + (10−1)·5 = 3 + 45 = 48.

  3. 3
    Step 3 of 4

    For the sum, use the sum formula

    Sₙ = n/2 · (2a₁ + (n−1)d). For n = 10: 10/2 · (2·3 + 9·5) = 5 · 51 = 255.

  4. 4
    Step 4 of 4

    Check the result

    Sanity check via the last term: a₁₀ = 48, the average of a₁ and a₁₀ is (3+48)/2 = 25.5, times 10 terms = 255 — checks out.

Examples

Arithmetic sequence — worked examples

nth term and sum with the full working
a₁=3, d=5, n=10 → aₙ
3 + (10−1)·5
3 + 45
48
a₁=3, d=5, n=10 → Sₙ
10/2 · (6 + 45)
5 · 51
255
a₁=2, d=3, n=20 → aₙ
2 + 19·3
2 + 57
59
1+2+…+100 (a₁=1, d=1)
100/2 · (2 + 99)
50 · 101
5050
a₁=10, d=−2, n=6 → aₙ
10 + 5·(−2)
10 − 10
0
a₁=5, d=5, n=8 → Sₙ
8/2 · (10 + 35)
4 · 45
180
Theory

What is an arithmetic sequence?

An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is always the same. This constant difference is called d. Given the first term a₁, every later term comes from adding d: a₂ = a₁ + d, a₃ = a₁ + 2d, and in general aₙ = a₁ + (n−1)·d. With positive d the sequence grows, with negative d it falls. The sum of the first n terms, Sₙ, follows from the famous Gauss trick: pairing the first term with the last, the second with the second-to-last, and so on, every pair has the same sum a₁ + aₙ. Hence Sₙ = n/2 · (a₁ + aₙ) = n/2 · (2a₁ + (n−1)d). Arithmetic sequences model steady, additive growth: instalment payments, rows of seats, stacks, uniform motion. They are the counterpart to the geometric sequence, where instead of adding you multiply by a constant factor.

Pitfalls

Common mistakes

Confusing (n−1) with n

The nth term adds (n−1)·d, not n·d. The first term (n=1) gets zero steps: a₁ = a₁ + 0·d.

Computing the difference the wrong way

d = a₂ − a₁, that is later minus earlier. For falling sequences d is negative.

Sum formula without the n/2

Sₙ = n/2 · (2a₁ + (n−1)d). The factor n/2 (half the number of terms) is often dropped.

Confusing arithmetic with geometric

Arithmetic sequences add d; geometric ones multiply by a ratio q.
FAQ

Frequently asked questions about arithmetic sequences

Glossary

Glossary — key terms explained simply

Arithmetic sequence
A number sequence with a constant difference between terms.
Term
A single value of the sequence; the nth is aₙ.
First term a₁
The starting value of the sequence.
Common difference d
The constant step: d = a₂ − a₁.
Partial sum Sₙ
The sum of the first n terms.
Gauss sum
The pairing idea behind the sum formula n/2 · (a₁ + aₙ).