Arithmetic Sequence Calculator — nth Term and Sum

An arithmetic sequence is a list of numbers where the difference between each term and the previous one is constant. This value is the common difference r. For example, 2, 5, 8, 11... is arithmetic with r = 3.

Use a_n = a₁ + (n − 1) · r, where a₁ is the first term, r the common difference and n the term number. For a₁ = 2, r = 3 the tenth term is a₁₀ = 2 + 9 · 3 = 29.

Use Sₙ = n · (a₁ + a_n) / 2 — the number of terms times the average of the first and last term. For a₁ = 2, a₁₀ = 29, n = 10: S₁₀ = 10 · 31 / 2 = 155.

It is the constant amount by which each term changes. Positive r means an increasing sequence, negative r a decreasing one, and r = 0 a constant sequence. You can find it as r = a_(n+1) − a_n.

In an arithmetic sequence terms are built by adding a constant difference (2, 5, 8, 11). In a geometric sequence they are built by multiplying by a constant ratio (2, 6, 18, 54). Arithmetic grows linearly, geometric exponentially.

Yes. A negative difference gives a decreasing sequence, e.g. for a₁ = 20, r = −3: 20, 17, 14, 11, 8... The term and sum formulas work the same regardless of the sign of r.

Saving a fixed amount each month, the number of seats in successive rows of an auditorium increasing by a fixed amount, instalments decreasing by an equal amount, or distance markers along a road.

If the first row has a₁ seats and each next row r more, then row n has a_n = a₁ + (n − 1) · r seats, and the total is Sₙ = n · (a₁ + a_n) / 2. E.g. 20 seats, +2 per row, 15 rows → 510 seats.

Yes, n is a position in the sequence, so a positive integer (1–100000); the calculator rounds down. The first term a₁ and difference r can be any value, including fractions and negatives.

Yes: Sₙ = n · (2a₁ + (n − 1) · r) / 2. It computes the sum directly from a₁, r and n without first finding a_n, giving the same result as the formula that uses the last term.