How this calculator works
Use this calculator to find the requested term, the sum of the first n terms, and the infinite sum when |r| < 1. Choose First term & common ratio or Two known terms. After a successful calculation, the result includes the formulas, calculation steps, derived values, and up to the first 50 sequence terms.
Inputs, formulas, and assumptions
In the first mode, enter First term (a₁), Common ratio (r), and Which term to solve for (n). The calculator uses:
a_n = a₁ * r^(n - 1)for the requested term.S_n = a₁ * (1 - r^n) / (1 - r)for the firstnterms, orS_n = a₁ * nwhenr = 1.S∞ = a₁ / (1 - r)when|r| < 1.
In Two known terms, enter each term's position and value. The calculator derives r from a_m = a_k * r^(m - k), then finds a₁. The positions must be different positive whole numbers, and known term values cannot be zero. The calculator assumes a constant ratio and does not test unprovided terms. It treats inputs as plain numbers; units are not inferred or converted.
Limits and accepted number formats
- Numeric fields accept ordinary signed decimal text, such as
-2or0.5. Scientific notation, fractions, comma-formatted numbers, and a leading plus sign are not accepted. - Positions and
nmust be positive whole numbers from 1 through1,000,000,000. - The calculator rejects a zero First term (a₁), zero Common ratio (r), or zero known-term value during calculation.
- When the position gap is even and the value ratio is positive, two real ratio choices can be possible. This implementation selects the positive root. It reports an error when an even root of a negative value would be required.
- First terms of the sequence shows at most 50 terms and adds a message when more terms are omitted.
- Very large powers or sums can exceed JavaScript's finite-number range, so valid inputs can still produce non-finite displayed results such as
InfinityorNaN. - Finite results and sequence terms are rounded for display, generally to no more than six decimal places. Calculations use JavaScript numeric values internally.
How to use the calculator
- Choose First term & common ratio or Two known terms.
- Enter the displayed values and Which term to solve for (n).
- Click Calculate. The result appears only after validation succeeds.
- Review the result, Formula & steps, Infinite sum when available, and First terms of the sequence.
- Use Copy result or Download result (.txt) to save the generated plain-text result.
Changing the selected mode clears the displayed result but does not automatically clear every field. Reset clears the fields and returns to First term & common ratio.
Privacy and page-level limitations
The sequence calculation runs in your browser, so the tool does not upload the calculator values. Separately, the page loads shared Ahrefs and Google Analytics resources, and the optional subscription form posts a submitted email to /assets/subscribe.php. The tool does not identify whether an existing list is geometric, handle complex ratios, infer units, or provide certified or professional calculations.
Frequently Asked Questions
What is a geometric sequence?
A geometric sequence is a list of numbers where each term after the first is found by multiplying the term before it by a fixed number, called the common ratio. For example, 3, 6, 12, 24 has a common ratio of 2.
What is the formula for the nth term?
The nth term is a_n = a₁ * r^(n - 1), where a₁ is the first term, r is the common ratio, and n is the term position.
What is the formula for the sum of a geometric sequence?
The sum of the first n terms is S_n = a₁ * (1 - r^n) / (1 - r) when r is not 1, or S_n = a₁ * n when r = 1. If |r| < 1, the infinite sum is S∞ = a₁ / (1 - r).
Can I calculate the sequence from two known terms?
Yes. Enter two nonzero term values and different positive whole-number positions. The calculator derives r from a_m = a_k * r^(m - k), then works backward to find a₁.
Does this calculator work with negative ratios or decimals?
Yes. The first term and common ratio can be positive, negative, or decimal values, but not zero. Enter ordinary signed decimal text; scientific notation, fractions, commas, and a leading plus sign are not accepted.
Why does the calculator use a positive root for some two-term inputs?
When the position gap is even and the value ratio is positive, both positive and negative real ratios can satisfy the relationship. The implementation chooses the positive root. It reports an error when no real ratio exists.
What are the calculator's position and display limits?
Positions and n can range from 1 through 1,000,000,000. First terms of the sequence displays at most 50 terms and indicates when more are omitted.
Can a valid calculation produce a non-finite result?
Yes. Very large powers or sums can exceed JavaScript's finite-number range and display a value such as Infinity or NaN.