Arithmetic Sequence Calculator

In an arithmetic sequence each term is the previous one plus the same amount, the common difference d: 3, 7, 11, 15… This calculator finds any term and the sum of the first n terms, writes the explicit and recursive formulas, and works backwards from any two known terms to the first term and the difference. It can also tell you whether a value appears in the sequence and where.

How to use this tool

1. Choose whether you know the first term and difference, or two terms and their positions.
2. Enter those values and the term number n you want.
3. Optionally enter a value to find its position in the sequence.
4. Read aₙ, Sₙ and the formulas; the table lists the first 20 terms with running sums.

Formula

aₙ = a₁ + (n − 1)d · Sₙ = n(a₁ + aₙ) ÷ 2 = n[2a₁ + (n − 1)d] ÷ 2 · from two terms: d = (aₖ − aₘ) ÷ (k − m)

The sum formula pairs the first and last terms, the second and second-to-last and so on: every pair adds to a₁ + aₙ, and there are n ÷ 2 pairs. The recursive form aₙ = aₙ₋₁ + d defines the same sequence one step at a time.

Example

For 3, 7, 11, … (a₁ = 3, d = 4), the 20th term is 3 + 19 × 4 = 79 and the first 20 terms add to 20 × (3 + 79) ÷ 2 = 820. Knowing only a₃ = 11 and a₁₀ = 39 gives the same sequence: d = (39 − 11) ÷ 7 = 4 and a₁ = 11 − 2 × 4 = 3.