Geometric Sequence Calculator

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio (r). For example, 2, 4, 8, 16, 32 is a geometric sequence with r=2.

The nth term is calculated using the formula: aₙ = a₁ · rⁿ⁻¹, where a₁ is the first term, r is the common ratio, and n is the term number.

The common ratio (r) is the constant multiplier between consecutive terms. It can be calculated as r = aₙ₊₁ / aₙ. It can be any non-zero real number.

For r ≠ 1, the partial sum is Sₙ = a₁ · (rⁿ - 1) / (r - 1). When r = 1, all terms are equal to a₁, so Sₙ = a₁ · n.

The infinite sum exists only when |r| < 1. It equals S = a₁ / (1 - r). When |r| ≥ 1, the series diverges and the infinite sum does not exist.

Yes, the common ratio can be negative. In this case, the terms alternate between positive and negative values. For example, with a₁=1 and r=-2, the sequence is: 1, -2, 4, -8, 16...

In an arithmetic sequence, a constant difference is added to each term (linear growth). In a geometric sequence, each term is multiplied by a constant ratio (exponential growth or decay).

Geometric sequences appear in compound interest, population growth, radioactive decay, depreciation of assets, music (equal temperament scale), and computer science (algorithm complexity).

A result of 0 for the infinite sum means |r| ≥ 1 and the geometric series diverges — the infinite sum mathematically does not exist, which is represented as 0 in this calculator.

The calculator supports 1 to 100 terms. For large n with |r| > 1, values can become very large. With |r| < 1, the partial sum asymptotically approaches the infinite sum.