Number Sequence Calculator

A sequence is a machine that emits the next term from a rule. You do not paste 2, 5, 8, 11 and ask for a mean. You say “start at 2, add 3 each time, give me the 5th term” and the machine returns 14, with the sum of those five terms equal to 40. That is the opposite of the statistics calculator, which only looks backward at numbers you already collected. If you already have the finished list and you want centre or spread, leave this page.

Arithmetic sequences add a constant d. The nth term is aₙ = a₁ + (n − 1)d. The sum of the first n terms is n/2 times (first + last), which is the same as n/2 × (2a₁ + (n − 1)d). Seats numbered 4, 7, 10, … or a savings plan that tucks away the same extra amount each month are arithmetic. d can be negative — a countdown that subtracts 5 still belongs here.

Geometric sequences multiply by a constant r. The nth term is aₙ = a₁ rⁿ⁻¹. 3, 6, 12, … has r = 2; the 4th term is 24. The sum is n × a₁ when r = 1 (every term equals the first). Otherwise Sₙ = a₁(rⁿ − 1)/(r − 1). Compounding a quantity by the same factor, or a shrinking half-life style series with |r| < 1, is geometric. r can be negative, which alternates sign.

n must be a positive integer. There is no “term number 2.5” and no zeroth term on this page. Huge n or a violent |r| can overflow; the calculator rejects Infinity instead of printing it. The geometric-sum formula shown in the details assumes the r ≠ 1 identity; when r is exactly 1 the implementation switches to n × a₁ so you still get a finite sum.

Nothing here estimates a mean, a standard deviation or a sample size. Those tools describe or plan data. This one only walks a declared arithmetic or geometric rule to a named position and totals the terms along the way.