DIY Cost Tools
Geometric Sequence Calculator
Find the nth term and finite sum of a geometric sequence using a first term, common ratio, and term count. Start with the example or enter your own numbers.
First term and ratio may be negative or zero. n must be an integer from 1 to 10,000. Input magnitudes must not exceed 1 trillion.
How to use
Enter the required numbers and select Calculate. Default results appear on page load. Invalid entries clear previous results and explain the problem. Answers display up to ten significant digits, so long results are rounded.
Formula and explanation
aₙ = a₁rⁿ⁻¹. For r ≠ 1, Sₙ = a₁(1 − rⁿ)/(1 − r); for r = 1, Sₙ = na₁. Every term after the first multiplies the preceding term by the common ratio r. Positive ratios use a numerically stable logarithm-based sum near r = 1.
Worked example
Starting at 3 with ratio 2 gives 3, 6, 12, 24, 48. The fifth term is 48 and the sum is 93.
Important limitations
Assumes an exactly constant ratio. This is a finite sequence tool, not an infinite-series convergence test. Very large powers may exceed numeric range and show an error; very small powers can underflow to zero. Floating-point results are estimates, not arbitrary-precision symbolic answers.
Frequently asked questions
What if the ratio is 1?
All terms equal the first term; the sum is n times that value.
Can the ratio be negative?
Yes. Signs alternate when the ratio is negative and the first term is nonzero.
What if the ratio is zero?
The first term remains, and all later terms are zero.
How is this different from an arithmetic sequence?
An arithmetic sequence adds a constant difference; a geometric sequence multiplies by a constant ratio.
Review notes
Reviewed October 1, 2026. Default results and boundary cases were independently checked. Validation was tested for blank fields, nonfinite numbers, excessive magnitudes, and invalid input formats or term counts. This is an educational calculator; verify precision requirements for professional use.