What is an arithmetic sequence?

A sequence where each term is the previous term plus a constant difference d. General term: aₙ = a₁ + (n−1)d.

What is a geometric sequence?

A sequence where each term is the previous term multiplied by a constant ratio r. General term: aₙ = a₁ × rⁿ⁻¹.

What is the sum to infinity of a geometric series?

If |r| < 1, the series converges to S∞ = a₁ / (1 − r). If |r| ≥ 1, the series diverges.

Number Sequence Calculator

Q: What is the sum to infinity of a geometric series?

If |r| < 1, the series converges to S∞ = a₁ / (1 − r). If |r| ≥ 1, the series diverges.

Enter the first term, common difference (arithmetic) or common ratio (geometric), and position n to compute the nth term and partial sum — plus the preview of the first terms.

How to use this calculator

Select arithmetic (constant difference) or geometric (constant ratio).
Enter the first term a₁ and the common difference d or ratio r.
Enter the term position n and read aₙ, the partial sum Sₙ, and a sequence preview.
For geometric sequences with |r| < 1, the sum to infinity S∞ is also shown.

Formula & method

nth term and partial sum formulas

Arithmetic: aₙ = a₁ + (n−1)d · Sₙ = n(a₁+aₙ)/2 · Geometric: aₙ = a₁ × rⁿ⁻¹ · Sₙ = a₁(1−rⁿ)/(1−r) · S∞ = a₁/(1−r) if |r|<1

a₁ = first term, d = common difference, r = common ratio, n = term position.

Example

Arithmetic a₁=2, d=3, n=10: a₁₀ = 2 + 9×3 = 29, S₁₀ = 10×(2+29)/2 = 155.

Key insights

Arithmetic sequences: each term adds a constant. Sequence 2, 5, 8, 11, … has d = 3.
Geometric sequences: each term multiplies by a constant. Sequence 2, 6, 18, 54, … has r = 3.
Gauss's trick: the sum of an arithmetic sequence from 1 to n is n(n+1)/2.
A geometric series with |r| < 1 converges to S∞ = a₁ / (1 − r).

How to interpret your result

Arithmetic vs geometric

Arithmetic sequences grow linearly (constant addition). Geometric sequences grow exponentially (constant multiplication).

Sum to infinity

Only converges for geometric sequences with |r| < 1. Example: 1 + 0.5 + 0.25 + … = 2.

Frequently asked questions

What is an arithmetic sequence?: A sequence where each term is the previous term plus a constant difference d. General term: aₙ = a₁ + (n−1)d.
What is a geometric sequence?: A sequence where each term is the previous term multiplied by a constant ratio r. General term: aₙ = a₁ × rⁿ⁻¹.
What is the sum to infinity of a geometric series?: If |r| < 1, the series converges to S∞ = a₁ / (1 − r). If |r| ≥ 1, the series diverges.

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