Geometric Series Calculator

A geometric series is the sum of the terms in a geometric sequence, where each term is obtained by multiplying the previous term by a fixed constant called the common ratio (r). For example, 2 + 6 + 18 + 54 is a geometric series with first term 2 and common ratio 3.

The formula for the finite sum of n terms is Sₙ = a₁(1 − rⁿ) / (1 − r) when r ≠ 1. If r = 1, then all terms are equal and Sₙ = n × a₁. Simply enter your first term, common ratio, and number of terms into the calculator to get the result.

An infinite geometric series converges only when the absolute value of the common ratio is less than 1 (|r| < 1). In that case, the sum is S∞ = a₁ / (1 − r). If |r| ≥ 1, the series diverges and has no finite sum.

The common ratio r is the constant factor by which each term is multiplied to get the next. You can find it by dividing any term by the term before it: r = a₂ / a₁. For instance, in the series 4, 12, 36, 108, the common ratio is 12 / 4 = 3.

An infinite geometric series converges (has a finite sum) when |r| < 1. It diverges (grows without bound or oscillates) when |r| ≥ 1. For example, the series 1 + 1/2 + 1/4 + 1/8 + … converges because r = 0.5, while 1 + 2 + 4 + 8 + … diverges because r = 2.

The nth term of a geometric sequence is given by aₙ = a₁ × r^(n−1), where a₁ is the first term, r is the common ratio, and n is the term index. For example, with a₁ = 3 and r = 2, the 5th term is 3 × 2⁴ = 48.

Yes, a geometric series can have a negative common ratio. This causes the terms to alternate between positive and negative values. An infinite series with a negative ratio can still converge as long as |r| < 1, for example r = −0.5.

A geometric sequence is an ordered list of numbers where each term is a fixed multiple of the previous one (e.g., 2, 6, 18, 54). A geometric series is the sum of those terms (e.g., 2 + 6 + 18 + 54 = 80). This calculator computes the series sum, not just the sequence.