Common Ratio Calculator
In a geometric sequence every term is the previous term multiplied by the same number, the common ratio. That single number unlocks everything else: the next terms, any term you name, the sum of the first n terms, and whether the sequence grows, fades, or flips sign. This calculator finds the ratio for you, three ways: paste a full sequence, enter two consecutive terms, or enter any two terms with their positions.
Quick Answer
Key takeaways
- The common ratio is always later term divided by earlier term. If the divisions do not all agree, the sequence is not geometric, and this calculator tells you instead of inventing a ratio.
- For terms that are not neighbors, divide the two terms and take the root of the position gap. An even gap can hide a second, negative ratio.
- The size of r decides the story: |r| under 1 shrinks toward zero and the infinite series converges; |r| over 1 explodes; a negative r flips the sign every term.
- Once r is known, a(n) = a(1) x r^(n-1) gives any term directly, with no need to write out the sequence.
Three ways to find the common ratio with this tool
Use the tab that matches what you have. Full Sequence suits a pasted list from a textbook or a spreadsheet: the tool checks every consecutive division before answering, so a sequence that is not truly geometric gets flagged rather than averaged into a fake ratio. Two Terms is the fastest route when you hold one pair of neighbors. Terms + Positions handles the exam-style problem where you know, say, the 1st and 3rd terms but not the one between them. The optional n field works with all three tabs and adds that term plus the sum of the first n terms to the results.
Divide neighbors. Pick any term and divide it by the term right before it: r = a(n+1) / a(n). One division is enough if you trust the sequence.
Check a second pair. Divide another pair. If the two quotients match, the sequence is geometric and that quotient is r. If they differ, it is not geometric.
Bridge a gap with a root. For terms k positions apart, r is the k-th root of (later term / earlier term). Watch the sign: an even gap admits a negative root as well.
Use r. Build a(n) = a(1) x r^(n-1) for any term, and S(n) = a(1) x (1 − r^n) / (1 − r) for sums.
The formulas, with the sign cases stated plainly
The common ratio is the constant multiplier between consecutive terms, and every other formula on this page is built from it.
These are the standard forms taught in algebra and precalculus sequences-and-series units, the same derivations Khan Academy’s finite geometric series derivation walks through step by step. Two sign cases deserve care. A negative ratio alternates the signs of the terms while the sizes follow |r|, which the behavior line in the results reflects. And when the position gap between two known terms is even, the root step produces a magnitude, not a signed answer: both +r and −r reproduce the two terms you entered. The calculator reports the positive root and says so in its working note; check the negative option by hand if your problem allows alternating signs.
Worked examples in all three modes
Each tab of the calculator is pre-filled with one of these cases, and the outputs below are exactly what the tool returns.
| Mode | Input | Result |
|---|---|---|
| Full Sequence | 2, 6, 18, 54, 162, with n = 8 | r = 3; next 5 terms 486, 1458, 4374, 13122, 39366; a(8) = 4374; S(8) = 6560; divergent |
| Two Terms | First term 4, next term 12 | r = 3 (12 / 4); general formula a(n) = 4 x 3^(n-1) |
| Terms + Positions | Term 1 = 5, term 3 = 20 | r = 2 from (20 / 5)^(1/2); first term recovered as 5 |
The sequence example shows why verification matters: 6/2, 18/6, 54/18, and 162/54 all equal 3, and the tool confirms that before reporting r. The positions example carries the subtlety flagged above: r = −2 fits the same two terms just as well (5, −10, 20), because the gap of 2 is even. Textbooks test exactly this point, and it is why the calculator’s note names the convention instead of hiding it. The same root-of-the-gap idea, run in reverse over years, is how a CAGR return calculator extracts a steady yearly growth rate from a start and end value.
What the value of r tells you at a glance
The ratio is a complete summary of a sequence’s long-run behavior, and the results card labels it using the same bands as this table.
| Value of r | What the terms do | Series verdict |
|---|---|---|
| 0 < r < 1 | Shrink toward zero, same sign | Converges S = a(1) / (1 − r) |
| r = 1 | Every term identical | Constant S(n) = n x a(1) |
| −1 < r < 0 | Shrink toward zero, signs alternate | Converges |
| r = −1 | Flips between a(1) and −a(1) forever | Alternates no sum |
| |r| > 1 | Grow without bound (alternating if r is negative) | Diverges |
| r = 0 | First term, then all zeros | Sum is just a(1) |
Growth language hides a ratio in everyday topics too. A quantity that grows by the same percentage each period is a geometric sequence wearing a suit: the ratio is 1 plus the growth rate. That is the same per-period multiplier an average growth factor calculator averages across uneven periods, and it is why spotting r in a list of values is a genuinely transferable skill, not just an exam trick.
What competing calculators miss
The best-known geometric sequence tools reviewed for this page, including Omni Calculator, DQYDJ, and Firgelli, are built backwards for this question: they ask you to type in the common ratio yourself, then generate terms and sums from it. If you already know r, they are fine tools. If finding r is the problem, they leave you where you started.
Miss 1: no ratio extraction from a sequence. None of the reviewed tools accepts a pasted sequence, divides the consecutive terms, and verifies that the quotients agree. This calculator does, and it refuses to answer when the quotients disagree, which is the honest result for a non-geometric list.
Miss 2: no positions mode. Recovering r from two terms with a gap between them is a standard textbook problem. The reviewed tools either skip it or fold it into a general two-unknown solver with the working buried. Here it is a dedicated tab with the root step shown in the results note.
Miss 3: behavior left implicit. Tools that print an infinite sum rarely state the convergence condition next to the number. The results card here names the behavior (convergent, constant, alternating, divergent) and only quotes an infinite sum when |r| is under 1.
One limit to know: results are rounded to six decimal places, so a ratio like 1/3 displays as 0.333333 and long generated terms can drift in the last digit. For exact fractional work, keep r as a fraction in your own algebra.
Frequently Asked Questions
Divide any term by the term immediately before it: r = a(n+1) / a(n). For 2, 6, 18, 54, that is 6/2 = 3, confirmed by 18/6 and 54/18. If the terms are not consecutive, divide them and take the root of the position gap.
Use the Terms + Positions tab: r = (later term / earlier term) raised to 1 over the position gap. With term 1 = 5 and term 3 = 20, r = (20/5)^(1/2) = 2. If the gap is even, the negative root also fits; the calculator reports the positive root and notes the convention.
Divide each term by the previous one. If every quotient is identical, the sequence is geometric and that quotient is r. If any quotient differs, it is not geometric. The Full Sequence tab performs this check for you and flags a mismatch instead of reporting a ratio.
The signs of the terms alternate: with a(1) = 5 and r = -2 the sequence runs 5, -10, 20, -40. The sizes still follow |r|, so the sequence converges if |r| is under 1 and diverges if |r| is over 1, exactly as with a positive ratio.
If r = 1, every term equals the first term and the sequence is constant, like 7, 7, 7, 7; the sum of n terms is n x a(1). If r = 0, every term after the first is 0, a valid but trivial case whose series sum is just the first term.
Only when |r| is under 1, so the terms shrink toward zero fast enough for the total to settle. Then S = a(1) / (1 – r). For 8, 4, 2, 1, … with r = 0.5, the infinite sum is 8 / 0.5 = 16. At |r| of 1 or more the series diverges and no finite sum exists.
A common ratio multiplies: each term is the previous term times r, giving a geometric sequence. A common difference adds: each term is the previous term plus d, giving an arithmetic sequence like 3, 7, 11, 15. Test which one your list follows by dividing first, then by subtracting.
Because at least one consecutive division differed from the others beyond rounding tolerance. Check for a typo in the pasted terms, a term copied out of order, or a list that is actually arithmetic. A sequence like 2, 6, 12, 20 fails because 6/2 = 3 but 12/6 = 2.
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References
- Khan Academy, “Finite Geometric Series Formula”: derivation of S(n) = a(1 − r^n) / (1 − r) used for the partial sums on this page. Khan Academy: Finite Geometric Series Formula. Accessed October 2026.
- Khan Academy, “Intro to Geometric Sequences”: the common ratio defined as the constant multiplier between consecutive terms, with the a(n) = a(1) x r^(n−1) form. Khan Academy: Intro to Geometric Sequences. Accessed October 2026.
- Khan Academy, “Worked Example: Finite Geometric Series (Sigma Notation)”: term-by-term verification of a finite sum against the formula. Khan Academy: Worked Example, Finite Geometric Series. Accessed October 2026.
- Khan Academy, “Geometric Series as a Function”: the convergence condition |r| under 1 behind the infinite-sum results quoted here. Khan Academy: Geometric Series as a Function. Accessed October 2026.
- Vedantu, “How Do You Find the Common Ratio of a Geometric Sequence”: the root-of-the-gap method for non-consecutive terms, r = (a(n) / a(m))^(1/(n−m)). Vedantu: Finding the Common Ratio of a Geometric Sequence. Accessed October 2026.