Geometric Sequence Calculator

A geometric sequence is an ordered list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). For example, 1, 2, 4, 8, 16 is a geometric sequence with a first term of 1 and a common ratio of 2.

The nth term is calculated using the explicit formula: aₙ = a₁ × r^(n−1), where a₁ is the first term, r is the common ratio, and n is the term position. For example, in the sequence 1, 2, 4, 8…, the 5th term is 1 × 2^(5−1) = 16.

The finite sum of n terms is Sₙ = a₁ × (1 − rⁿ) / (1 − r) when r ≠ 1. If r = 1, all terms are equal so Sₙ = n × a₁. This formula adds up all terms from the first to the nth term of the sequence.

An infinite geometric series is the sum of all terms in an infinite geometric sequence. It only converges (has a finite value) when the absolute value of the common ratio is less than 1 (|r| < 1). In that case, S∞ = a₁ / (1 − r). If |r| ≥ 1, the series diverges to infinity.

The common ratio is the constant multiplier between consecutive terms. To find it, divide any term by the term before it: r = aₙ / aₙ₋₁. For example, in the sequence 3, 9, 27, 81, the common ratio is 9 ÷ 3 = 3.

A geometric sequence is simply the ordered list of terms (e.g., 1, 2, 4, 8, 16), while a geometric series is the sum of those terms (e.g., 1 + 2 + 4 + 8 + 16 = 31). This calculator computes both the individual terms and the series sum.

Yes. A negative common ratio produces an alternating sequence (e.g., 1, −2, 4, −8). A fractional ratio between −1 and 1 (excluding 0) creates a decreasing sequence and the infinite series will converge. A ratio of exactly 0 is not allowed as it would make all terms after the first equal to zero.

The recursive formula defines each term using the previous one: a₁ = first term, and aₙ = aₙ₋₁ × r for n > 1. While the explicit formula aₙ = a₁ × r^(n−1) is more efficient for finding a specific term directly, the recursive approach builds the sequence step by step.