The most advanced geometric sequence calculator online free — find the nth term, common ratio, partial sum, and infinite series sum with complete step-by-step solutions. Works without the common ratio, supports fractions, generates full sequence tables, verifies sequences, and shows every step of the solution — just like Mathway but completely free.
Choose a mode, enter values (numbers or fractions), get a complete step-by-step solution instantly
Seven modes, step-by-step solutions, built-in examples, fraction support, visual charts, and every formula — entirely free.
Find any term using aₙ = a₁·rⁿ⁻¹. The geometric sequence calculator nth term shows every substitution and power calculation with a full geometric sequence calculator with solution output.
Derive r from the first term and any known term using the geometric sequence ratio formula r = (aₙ/a₁)^(1/(n−1)). Full step-by-step working with verification.
Applies the geometric sequence sum formula Sₙ = a₁(1−rⁿ)/(1−r) with every step shown. Supports partial sums from any starting term using the subtraction geometric sequence calculator technique.
Calculates S∞ = a₁/(1−r) when |r|<1. Checks convergence first, shows partial sums approaching the limit, and warns when the geometric sequence sum diverges.
The geometric sequence calculator without common ratio derives r and a₁ from two known terms and their positions — the most tested exam technique and the top-searched geometric sequence calculator use case.
Full geometric sequence fraction calculator support — enter terms and ratios as fractions like 1/2 or 3/4. All steps display exact decimal results with full precision throughout.
Paste any comma-separated list of numbers to check if they form a geometric sequence. Confirms constant ratio, finds a₁ and r, and applies the geometric sequence sum formula automatically.
Six classic geometric sequence examples — doubling, halving, fractions, negative ratio, and more — pre-loaded and clickable to instantly populate any calculator mode.
Every geometric sequence calculator with steps mode shows numbered, colour-coded solution steps. Exactly like a geometric sequence calculator Mathway-style tool — completely free.
An auto-scaled SVG chart visualises geometric growth or decay of the sequence terms — instantly showing whether the pattern is exponential growth, convergent decay, or oscillation.
Generate a complete table with every term value, ratio check, and cumulative sum for up to 25 terms — the most detailed geometric calculator online table available free.
This geometric sequence calculator online free runs entirely in your browser. No account, no subscription, no data sent anywhere. Use for homework, exams, teaching, or professional work.
The complete reference for using the geometric sequence calculator with steps — from basic definitions to advanced techniques for exams and professional applications
A geometric sequence is a list of numbers in which each term after the first is found by multiplying the preceding term by a fixed, non-zero constant — the common ratio. The word "geometric" reflects the multiplicative relationship between consecutive terms, in contrast to an arithmetic sequence where the relationship is additive. Using a geometric sequence calculator makes it easy to work with sequences of any complexity without tedious manual computation.
The general form of any geometric sequence is: a₁, a₁r, a₁r², a₁r³, …, a₁rⁿ⁻¹, where a₁ is the first term, r is the common ratio, and n is the term number. The geometric sequence calculator online free tool above handles all six modes of geometric sequence calculation — from finding a specific term to computing infinite sums — with full step-by-step working shown at every stage.
Here are six essential geometric sequence examples that any complete geometric calculator online should handle:
These geometric sequence examples represent the full range of behaviour the geometric sequence calculator with steps must handle — and all are pre-loaded as one-click examples in the tool above.
The geometric sequence ratio formula in its simplest form is: r = aₙ / aₙ₋₁ — any term divided by the term immediately before it. This is the foundation of the geometric sequence calculator common ratio mode: given consecutive terms, the ratio is immediate.
The more powerful version of the geometric sequence ratio formula applies between any two non-consecutive terms at positions m and n: r = (aₙ / aₘ)^(1/(n−m)). This extended ratio formula is the core of the geometric sequence calculator without common ratio technique — the ability to determine r from any two data points in the sequence, regardless of their positions. The geometric sequence calculator with steps shows the complete root derivation for this formula, making it clear why the exponent 1/(n−m) represents the "average multiplicative factor per step" between the two known terms.
Once the geometric sequence ratio formula yields r, everything else follows: the first term a₁, all subsequent terms via the nth term formula, and the partial and infinite sums via the geometric sequence sum formula. The geometric sequence calculator common ratio output feeds directly into all these downstream calculations.
The geometric sequence ratio formula is also the foundation of the Verify Sequence mode: if every consecutive pair of terms in a list has the same ratio, the list is confirmed to be a geometric sequence. The geometric calculator online verify tool automates all these ratio checks and reports the result instantly.
The nth term geometric sequence calculator implements the formula aₙ = a₁ · rⁿ⁻¹. This formula is arguably the most important single result in geometric sequence theory — it allows direct computation of any term without iterating through every intermediate value.
The geometric sequence calculator nth term technique proceeds in four steps, each of which the geometric sequence calculator with steps displays explicitly:
The most common error students make with the geometric sequence calculator nth term formula is using n instead of n−1 as the exponent. The first term a₁ corresponds to n = 1, which gives r⁰ = 1 — the exponent is always one less than the term position. The geometric sequence calculator with solution output highlights this exponent calculation with a dedicated step to prevent this mistake.
This step-by-step approach is exactly the geometric sequence calculator technique used by tools like geometric sequence calculator Mathway — but here it is provided completely free. Students using this geometric sequence calculator online free tool get the same educational value as a paid subscription service.
The geometric sequence sum formula is one of the most powerful and widely applicable results in mathematics. For the sum of the first n terms of a geometric sequence with first term a₁ and common ratio r ≠ 1:
Sₙ = a₁ · (1 − rⁿ) / (1 − r)
When r = 1, the geometric sequence sum formula degenerates to Sₙ = n · a₁ (all terms are equal). The geometric sequence sum calculator above handles both cases automatically, displaying the appropriate formula variant in the step-by-step solution.
The geometric sequence sum calculator breaks this formula into four sub-steps in the solution output: compute rⁿ; compute the numerator (1 − rⁿ); compute the denominator (1 − r); divide and multiply by a₁. This decomposition prevents the most common calculation errors, which occur when students try to apply the entire formula in one step without tracking intermediate values.
One of the most powerful geometric sequence calculator techniques for partial sums is the subtraction approach: the sum from term m to term n is simply Sₙ − Sₘ₋₁. This means you compute the full sum of the first n terms, then subtract the sum of the first (m−1) terms. The geometric sequence sum calculator with the "sum from" input implements this automatically, showing both sub-sums before computing the difference.
This technique is essential for exam questions of the form "find the sum of terms 4 to 10" — a question that is simply S₁₀ − S₃ using the geometric sequence sum formula. The geometric sequence calculator with steps displays both components of this subtraction explicitly, giving a complete geometric sequence calculator with solution that is easy to follow and reproduce.
The infinite geometric sequence sum formula is: S∞ = a₁ / (1 − r), valid only when |r| < 1. This elegant result shows that an infinite number of terms can sum to a finite value — one of the most surprising discoveries in the history of mathematics. The series 1 + 1/2 + 1/4 + 1/8 + … sums to 2, not infinity, because each term is smaller than the last in the right proportion.
The infinite sum mode of this geometric calculator online checks the convergence condition |r| < 1 before computing, warns explicitly when the series diverges, and verifies the result by showing several partial sums that approach the infinite limit. This convergence check is a required step in any complete geometric sequence calculator with solution for infinite series.
The geometric sequence calculator without common ratio mode is the most commonly searched and most frequently exam-tested feature of any geometric sequence tool. It addresses the scenario where you are given two terms at known positions but the common ratio is not stated — and you must derive everything from that limited information.
The geometric sequence calculator without common ratio derivation uses the extended ratio formula: given that a_m and a_n are known, r = (aₙ / aₘ)^(1/(n−m)). Once r is found, a₁ is back-calculated as a₁ = aₘ / rᵐ⁻¹. From a₁ and r, all other terms and sums follow using the standard formulas.
For example: if you know that the 3rd term is 12 and the 7th term is 192, then r = (192/12)^(1/4) = 16^(0.25) = 2, and a₁ = 12 / 2² = 3. The geometric sequence calculator without common ratio mode in this tool shows every step of this derivation labelled and explained, producing a complete geometric sequence calculator with solution without any assumed starting knowledge of r.
This technique is widely discussed in geometric sequence calculator Reddit threads and taught in evidence-based programs like the approach used by geometric sequence calculator built with science-style education platforms. It is tested on SAT, ACT, IB, A-Level, GCSE, and university entrance exams worldwide. This free geometric sequence calculator with steps provides the same quality of guided solution as a geometric sequence calculator Mathway subscription — at zero cost.
The geometric sequence fraction calculator capability matters because many real-world geometric sequences involve fractional ratios. Financial discount factors, probability of repeated events, radioactive decay, and geometric optics all generate sequences with fractional common ratios. The geometric sequence fraction calculator accepts any fraction in the form a/b in the input fields and carries full decimal precision through every calculation step.
| Mode | Formula | Inputs Needed | Main Output |
|---|---|---|---|
| nth Term | aₙ = a₁ · rⁿ⁻¹ | a₁, r, n | Any specific term |
| Find Ratio | r = (aₙ/a₁)^(1/(n−1)) | a₁, aₙ, n | Common ratio r |
| Partial Sum | Sₙ = a₁(1−rⁿ)/(1−r) | a₁, r, n | Sum of n terms |
| Infinite Sum | S∞ = a₁/(1−r), |r|<1 | a₁, r | Infinite series sum |
| Without Ratio | r = (aₙ/aₘ)^(1/(n−m)) | Two terms + positions | a₁, r, any aₙ, Sₙ |
| Full Sequence | aₙ = a₁ · rⁿ⁻¹ (repeated) | a₁, r, count | Complete table + chart |
| Verify | Check ratio = aₙ/aₙ₋₁ | List of terms | Is geometric? a₁, r, Sₙ |