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Arithmetic & Geometric Sequence Calculator

Find the nth term and the sum of the first n terms of an arithmetic or geometric sequence.

Term #10
48
Sum of first 10 terms
255

How it works

An arithmetic sequence adds the same amount each step: the nth term is a₁ + (n−1)d. Its sum of the first n terms uses Gauss's pairing trick — Sₙ = n/2 × (2a₁ + (n−1)d), the average of the first and last term, times how many terms there are.

A geometric sequence multiplies by the same ratio each step: the nth term is a₁ × r^(n−1). Its sum of the first n terms is Sₙ = a₁ × (rⁿ − 1) / (r − 1), except when r = 1, where every term equals a₁ and the sum is just n × a₁ (the general formula's denominator would otherwise be zero).

The term number n must be a whole number of at least 1 — a 'zeroth' or fractional term isn't defined for either kind of sequence.

FAQ

What's the difference between a common difference and a common ratio?

A common difference (arithmetic) is added to get the next term — sequences grow or shrink at a constant rate, like 3, 8, 13, 18. A common ratio (geometric) is multiplied to get the next term — sequences grow or shrink exponentially, like 2, 6, 18, 54. The same 'step' input means different things depending on which mode you pick.

Can the common ratio be negative or a fraction?

Yes — a negative ratio makes the sequence alternate sign each term (e.g. 2, −6, 18, −54), and a ratio between −1 and 1 (excluding 0) makes each term shrink toward zero rather than grow. Both are valid geometric sequences and this calculator handles them the same way.

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