Arithmetic and Geometric Sequence Calculator

An arithmetic sequence adds the same amount, the common difference d, to get from one term to the next, so the gaps are constant. A geometric sequence multiplies by the same amount, the common ratio r, so each term is a fixed multiple of the one before. Arithmetic sequences grow in a straight line, while geometric sequences grow or shrink exponentially.

For an arithmetic sequence the nth term is a1 + (n - 1)d and the sum of the first n terms is n/2 times (2*a1 + (n - 1)d). For a geometric sequence the nth term is a1 * r^(n-1) and the sum is a1 * (r^n - 1) / (r - 1), or simply a1 * n when r is exactly 1. This calculator applies these formulas and also lists every term so you can check the pattern.

Yes, but only when the common ratio satisfies |r| < 1, so the terms keep shrinking toward zero. In that case the infinite sum converges to a1 / (1 - r). When |r| is 1 or more the terms do not shrink, the running total keeps growing, and the infinite sum does not converge to a finite value.

The first term, the common difference and the common ratio can each be any finite number, including negatives and decimals. A negative difference makes an arithmetic sequence count down, and a negative ratio makes a geometric sequence alternate between positive and negative terms. The number of terms n must be a whole number of at least 1.

No. Every calculation runs entirely in your browser, so the first term, the difference or ratio and your results never leave your device.