Arithmetic Sequence Calculator

How it works

An arithmetic sequence adds the same amount, the common difference d, to each term to get the next: 2, 5, 8, 11, … has a₁ = 2 and d = 3. Because every step is the same size, the nth term is the first term plus n − 1 steps: aₙ = a₁ + (n − 1)d. A negative d gives a decreasing sequence.

The sum of the first n terms, written Sₙ, has a shortcut Gauss is said to have found as a schoolboy: pair the first term with the last, the second with the second-to-last, and so on. Every pair adds to a₁ + aₙ and there are n ÷ 2 pairs, so Sₙ = n(a₁ + aₙ) ÷ 2, the number of terms times the average of the first and last.

Any one of a₁, d, n and aₙ can be found from the other three by rearranging the nth-term formula. Choose what to find; the calculator reports the full set, the sum, the explicit formula and the first terms with their running sums so you can see the pattern.

Formula

aₙ = a₁ + (n − 1)d
Sₙ = n(a₁ + aₙ) ÷ 2 = n[2a₁ + (n − 1)d] ÷ 2
d  = (aₙ − a₁) ÷ (n − 1)
a₁ = aₙ − (n − 1)d
n  = (aₙ − a₁) ÷ d + 1

Example

a₁ = 2, d = 3, n = 10: a₁₀ = 2 + 9 × 3 = 29, and S₁₀ = 10 × (2 + 29) ÷ 2 = 155. The sequence runs 2, 5, 8, 11, 14, 17, 20, 23, 26, 29.

The classic 1 + 2 + 3 + … + 100 is a₁ = 1, d = 1, n = 100: a₁₀₀ = 100 and S₁₀₀ = 100 × 101 ÷ 2 = 5,050.

Which term of 3, 7, 11, … is 99? n = (99 − 3) ÷ 4 + 1 = 25. The sum of those 25 terms is 25 × (3 + 99) ÷ 2 = 1,275.

Assumptions and limitations

• n is a whole number of at least 1. Finding n only succeeds when the value you enter is actually a term of the sequence; otherwise the calculation reports that it is not.
• Finding the common difference needs at least two terms (n ≥ 2), and finding n needs a non-zero difference, since a constant sequence has every value at every position.
• The table lists at most the first 25 terms. The nth term and the sum are computed from the closed formulas, so n can be as large as you like.
• Results are cleaned to 15 significant digits and shown to 4 decimal places.

Frequently asked questions