Geometric Sequence Calculator

A list of numbers where each term is found by multiplying the previous one by a fixed number, the common ratio. 3, 6, 12, 24 is geometric with a ratio of 2.

It is the constant multiplier between consecutive terms, written r. You find it by dividing any term by the one before it: in 5, 15, 45 the ratio is 15 ÷ 5 = 3.

aₙ = a₁·r^(n−1), where a₁ is the first term, r the common ratio and n the position. The exponent is n−1 because the first term needs no multiplication.

Sₙ = a₁·(rⁿ − 1) ÷ (r − 1) for any ratio other than 1. When r is exactly 1 every term equals a₁, so the sum is simply a₁ × n.

A geometric sequence multiplies by a fixed ratio each step; an arithmetic sequence adds a fixed amount. 2, 6, 18 is geometric (×3); 2, 5, 8 is arithmetic (+3).

Yes. A ratio between −1 and 1 makes the terms shrink toward zero, and a negative ratio makes them alternate between positive and negative. Both work here.

When the ratio’s size is less than 1, the infinite sum settles on a finite value, a₁ ÷ (1 − r). For a₁ = 1 and r = 0.5 that limit is 2, which the partial sums approach as you add terms.

A ratio bigger than 1 raised to a high power grows past the largest number the browser can represent (about 1.8 × 10³⁰⁸). Rather than display infinity, the tool tells you the result is out of range.

No. The sequence is generated and summed entirely in your browser; nothing you type is sent anywhere.