In an AP we add a constant difference to each term (linear growth). In a GP we multiply by a constant ratio (exponential growth). AP: 2, 5, 8, 11… (d = 3). GP: 2, 6, 18, 54… (r = 3).
Carregando ferramenta… / Loading tool…
Arithmetic and geometric progressions: general term, sum and term list.
A progression is a sequence with a fixed pattern between neighboring terms. In an arithmetic progression (AP), each term is the previous one plus a constant difference d, giving linear growth; the general term is aₙ = a₁ + (n − 1)·d and the sum of the first n terms is Sₙ = n·(a₁ + aₙ) ÷ 2. In a geometric progression (GP), each term is the previous one times a constant ratio r, giving exponential growth (or decay); the general term is aₙ = a₁·rⁿ⁻¹ and the sum is Sₙ = a₁·(rⁿ − 1) ÷ (r − 1) when r ≠ 1 (if r = 1, all terms are equal and Sₙ = a₁·n). The tool generates from 1 up to 1000 terms, computes the nth term and the partial sum, and shows the step-by-step with the general term. The ratio can be an integer, a decimal or negative (period or comma as the separator). Everything runs locally in your browser with 64-bit floating point.
The essential difference is what repeats at each step. In an AP an addition repeats: you start at a₁ and add d each term, so the nth term is aₙ = a₁ + (n − 1)·d, a straight line. In a GP a multiplication repeats: you start at a₁ and multiply by r each term, so aₙ = a₁·rⁿ⁻¹, an exponential curve.
The sums follow the same spirit. The sum of an AP is Sₙ = n·(a₁ + aₙ) ÷ 2, the number of terms times the average of the first and last. The sum of a GP is Sₙ = a₁·(rⁿ − 1) ÷ (r − 1) when r ≠ 1; in the special case r = 1 the GP becomes a constant sequence and the sum is simply a₁·n.
Example 1, AP with a₁ = 3, d = 5, 6 terms: the sequence is 3, 8, 13, 18, 23, 28. The sixth term is a₆ = 3 + (6 − 1)·5 = 3 + 25 = 28, and the sum is S₆ = 6·(3 + 28) ÷ 2 = 6·31 ÷ 2 = 93.
Example 2, GP with a₁ = 2, r = 3, 4 terms: the sequence is 2, 6, 18, 54. The fourth term is a₄ = 2·3⁴⁻¹ = 2·27 = 54, and the sum is S₄ = 2·(3⁴ − 1) ÷ (3 − 1) = 2·80 ÷ 2 = 80 (check: 2 + 6 + 18 + 54 = 80).
The ratio shapes the sequence. In a GP, a negative r alternates the sign of the terms (2, −6, 18, −54…); an r between −1 and 1 makes the sequence tend to zero (the decreasing case); and r > 1 blows the values up quickly. For GPs with a large ratio and many terms, the numbers can overflow floating-point precision, hence the 1000-term limit and the care needed with huge values.
The two progressions model different phenomena. An AP describes linear growth: fixed loan installments, theater seats increasing row by row, constant depreciation. A GP describes exponential growth (or decay): compound interest, population growth, radioactive half-life and an asset that loses a fixed fraction of its value each year.
Paste the code into your HTML and the tool shows up on your page, without J-Kit's navigation and ads. It still runs in the browser of whoever visits your site.
<iframe
src="https://jkit.tools/embed/en-US/arithmetic-geometric-sequence-calculator"
width="100%"
height="600"
style="border:0"
loading="lazy"
title="Arithmetic & Geometric Sequence Calculator"
></iframe>These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.
In an AP we add a constant difference to each term (linear growth). In a GP we multiply by a constant ratio (exponential growth). AP: 2, 5, 8, 11… (d = 3). GP: 2, 6, 18, 54… (r = 3).
Calculations run in your browser. No data is sent to a server.