Math · Sequences
Geometric sequence calculator.
Give the first term and common ratio, or any two terms. You get the nth term, the sum of the first n terms, whether the infinite series converges, and the steps.
For a first term of 2 and a common ratio of 3, the 6th term is 2 × 3⁵ = 486 and the sum of the first 6 terms is 2(1 − 3⁶) / (1 − 3) = 728. The infinite series diverges because |r| is not less than 1.
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The formulas
A geometric sequence multiplies by the same common ratio r each step. The nth term is aₙ = a₁rⁿ⁻¹. The sum of the first n terms is Sₙ = a₁(1 − rⁿ) / (1 − r) for r ≠ 1, and n × a₁ when r = 1. An infinite geometric series has a finite sum only when −1 < r < 1, in which case the sum is a₁ / (1 − r); otherwise it diverges.
Worked example
a₁ = 2, r = 3, n = 6: the terms are 2, 6, 18, 54, 162, 486, so a₆ = 486 and S₆ = 2(1 − 729) / (1 − 3) = 728. With a₁ = 1, r = 0.5 and n = 10, a₁₀ = 0.001953 and S₁₀ = 1.998047, and because |r| < 1 the infinite series converges to 1 / (1 − 0.5) = 2. From two terms, a₂ = 6 and a₅ = 162 give r = (162 / 6)^(1/3) = 3 and a₁ = 2, so a₇ = 1,458.
Limits
From two terms the ratio is a real root, so terms of opposite sign an even number of steps apart have no real ratio. n is limited to 10,000 and very large results show in exponent form.
Sources, opened 15 October 2026: OpenStax, Algebra and Trigonometry 2e, 13.3 Geometric Sequences (common ratio; explicit formula aₙ = a₁rⁿ⁻¹); OpenStax, Algebra and Trigonometry 2e, 13.4 Series and Their Notations (sum of the first n terms Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1; infinite sum defined when −1 < r < 1).
Frequently asked questions
How do you find the nth term of a geometric sequence?
Use aₙ = a₁rⁿ⁻¹. With a₁ = 2 and r = 3, the 6th term is 2 × 3⁵ = 486.
How do you find the sum of a geometric series?
For r ≠ 1, Sₙ = a₁(1 − rⁿ) / (1 − r). For a₁ = 2, r = 3 and n = 6 the sum is 2(1 − 729) / (1 − 3) = 728.
When does an infinite geometric series converge?
Only when the absolute value of the common ratio is less than 1. Then the sum is a₁ / (1 − r). With a₁ = 1 and r = 0.5 the sum is 2.
How do I find the common ratio from two terms?
Divide the later term by the earlier one and take the root of the number of steps between them: r = (a_q / aₚ)^(1/(q − p)). For a₂ = 6 and a₅ = 162, r = 27^(1/3) = 3.
What if the ratio is negative?
The terms alternate in sign. The formulas still work, and the series converges if the ratio is between −1 and 0.
Learning aid
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