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Algebra · Sequences

Arithmetic & Geometric Progression Calculator

Find a selected term and the finite partial sum of an arithmetic or geometric progression, with the recurrence pattern and substitution explained.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Finite arithmetic and geometric sequences indexed from the first term

Arithmetic term

aₙ = a₁ + (n−1)d

Arithmetic sum

Sₙ = n(a₁+aₙ)/2

Geometric model

aₙ = a₁rⁿ⁻¹Sₙ = a₁(1−rⁿ)/(1−r)
Algebra · Sequences

Find a term and its partial sum

Inputs and displayed decimal results use at most three decimal places. Exact internal values are retained during calculation.

aₙ = a₁ + (n−1)d

Choose a progression and enter its defining values.

Using the model

Symbols, assumptions and limitations

Interpretation notes

The index n is a positive whole number. Arithmetic progressions add a constant difference d; geometric progressions multiply by a constant ratio r. Results retain full internal precision and display at most three decimals.

Worked examples

See the method in practice

01

Arithmetic sequence

For a₁ = 3, d = 2, and n = 5, the fifth term is 11 and the first five terms sum to 35.

02

Geometric sequence

For a₁ = 2, r = 3, and n = 4, the fourth term is 54 and the partial sum is 80.

Questions

Frequently asked

What is the difference between d and r?

d is added between arithmetic terms, while r multiplies consecutive geometric terms.

What happens when r equals 1?

Every geometric term equals the first term, so the partial sum is n times a₁.

Can the ratio be negative?

Yes. A negative ratio makes term signs alternate.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax College Algebra — Sequences

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology