Sequence Calculator — Arithmetic & Geometric
Find the nth term and the sum of the first n terms for an arithmetic (add a constant) or geometric (multiply by a constant) sequence. Enter the first term, common difference or ratio, and how many terms to include.
Sequence type
aₙ = a₁ + (n − 1) × d
- 1
Step multiplier (n − 1)
10 − 1 = 9 - 2
Increment (n − 1) × d
9 × 3 = 27 - 3
nth term = a₁ + (n − 1) × d
2 + 27 = 29Adds the total increment to the first term.
How does this calculator work?
Arithmetic: nth term = a₁ + (n−1)d, sum = n/2 × (2a₁ + (n−1)d). Geometric: nth term = a₁ × rⁿ⁻¹, sum = a₁(1−rⁿ)/(1−r) for r ≠ 1. Enter first term, common difference (arithmetic) or ratio (geometric), and n to get the nth term and partial sum.
Formula
How this is calculated
An arithmetic sequence adds the same constant d (the common difference) to each term: 2, 5, 8, 11, … (d = 3). The nth term is aₙ = a₁ + (n−1)d. The sum of the first n terms is Sₙ = n/2 × (2a₁ + (n−1)d), which is equivalent to n times the average of the first and last terms.
A geometric sequence multiplies each term by the same constant r (the common ratio): 3, 6, 12, 24, … (r = 2). The nth term is aₙ = a₁ × r^(n−1). The sum is Sₙ = a₁ × (1 − rⁿ) / (1 − r) for r ≠ 1, or simply n × a₁ when r = 1 (all terms equal). For |r| < 1 the infinite sum converges to a₁ / (1 − r), but this calculator computes only finite partial sums.
Edge cases: a geometric sequence with r = 0 has only a non-zero first term, so the sum equals a₁ alone. The calculator is limited to n ≤ 100 to keep outputs readable; for very large r, terms grow exponentially and may overflow to infinity.
Frequently asked questions
A sequence is an ordered list of terms (e.g., 2, 5, 8, 11). A series is the sum of those terms (e.g., 2 + 5 + 8 + 11 = 26). This calculator gives both the nth term of the sequence and Sₙ, the series (partial sum).
A geometric series converges (has a finite infinite sum) only when the common ratio satisfies |r| < 1. When |r| ≥ 1, the partial sums grow without bound. The calculator shows the finite partial sum Sₙ; an infinite geometric series sum would require |r| < 1.
Yes. A negative common difference gives a decreasing arithmetic sequence; a negative ratio gives an alternating geometric sequence. Fractional ratios (0 < r < 1) produce sequences that shrink toward zero. All are supported.
Also known as
TG we-Calculate Editorial Team. (2026). Sequence Calculator — Arithmetic & Geometric [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sequence-calculator
TG we-Calculate Editorial Team. "Sequence Calculator — Arithmetic & Geometric." TG we-Calculate. 2026. https://we-calculate.com/calculator/sequence-calculator.
TG we-Calculate Editorial Team, "Sequence Calculator — Arithmetic & Geometric," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sequence-calculator
@misc{wecalculate_sequence_calculator, title = {Sequence Calculator — Arithmetic & Geometric}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sequence-calculator}}, year = {2026}, note = {TG we-Calculate} }
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