Common Ratio Calculator: Find the Ratio

This calculator finds the common ratio of a geometric sequence. You enter two consecutive terms (or any two terms and their positions), and it returns the common ratio, along with the general term formula and the next term. The common ratio is the constant factor you multiply by to get from one term to the next.

Formula

The common ratio r of a geometric sequence is the quotient of any term divided by the previous term:

r = a_n / a_n−1

where:

• a_n is the n-th term (any term after the first)
• a_n−1 is the term immediately before it

If you only know two non-consecutive terms, use the more general form:

r = (a_m / a_k)^{1/(m−k)}

where a_m and a_k are the m-th and k-th terms. This is the formula the calculator uses when you provide two terms and their positions.

Worked Example

Using the calculator defaults: first term = 2, second term = 6.

1. Divide the second term by the first: 6 ÷ 2 = 3.
2. The common ratio is 3.
3. The next term is 6 × 3 = 18.
4. The general term is a_n = 2 × 3^(n−1).

The calculator’s first output is 3, matching the step-by-step result.

What Is a Geometric Sequence?

A geometric sequence (or geometric progression) is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, 5, 10, 20, 40 has a common ratio of 2. If the ratio is between −1 and 1 (e.g., 0.5), the terms get smaller and approach zero. If the ratio is negative (e.g., −2), the terms alternate in sign: 3, −6, 12, −24.

The common ratio is what makes the sequence “geometric� — it is the constant multiplier that defines the pattern. Knowing the ratio lets you predict any term, sum the sequence, or model exponential growth and decay.

When to Use a Common Ratio Calculator

You would use this calculator in any situation where a quantity changes by a constant factor each step:

• Math homework: finding the ratio from given terms of a geometric progression.
• Finance: compound interest, depreciation, or inflation where a value multiplies by a fixed rate each period.
• Biology: population growth where the number of organisms multiplies by a constant factor each generation.
• Physics: radioactive decay or signal attenuation, where the amount decreases by a constant fraction per unit time.
• Computer science: analyzing algorithms that double in size each step.

In all these cases, the common ratio is the “growth factor� — if it’s greater than 1, the sequence grows; if it’s less than 1, it decays.

How to Read the Result

The common ratio is a pure number with no units. If the ratio is:

• Greater than 1: the sequence increases (e.g., 2, 4, 8).
• Between 0 and 1: the sequence decreases but stays positive (e.g., 100, 50, 25).
• Negative: the terms alternate between positive and negative (e.g., 1, −3, 9).
• Exactly 1: all terms are equal (e.g., 7, 7, 7).
• Exactly −1: the sequence alternates between two values (e.g., 4, −4, 4).

The calculator also shows the next term and the general formula, which are useful for continuing the sequence or finding any term directly.

Common Mistakes and Pitfalls

• Using the wrong order: the ratio is always later term divided by earlier term. Dividing the first term by the second gives the reciprocal, which is wrong.
• Mixing up arithmetic and geometric sequences: an arithmetic sequence adds a constant difference, not multiplies. For example, 2, 5, 8 has a common difference of 3, not a common ratio.
• Forgetting to account for non-consecutive terms: if you have the 2nd term and the 5th term, you can’t just divide them — you must take the cube root (because 5 − 2 = 3). The calculator’s general formula handles this if you enter the positions.
• Rounding too early: if the ratio is irrational (like √2), rounding to a few decimals can change the next term noticeably. Keep full precision until the final step.
• Assuming a zero term: a geometric sequence cannot have a zero term (except the all-zero sequence), because division by zero is undefined. If you see a zero term, the sequence is not geometric.

Limits of the Formula

The common ratio formula works only for geometric sequences. If the sequence is not geometric (e.g., 1, 4, 9, 16), the ratio between consecutive terms is not constant, and this calculator will give a meaningless result. Also, the formula assumes the terms are real numbers; complex ratios are possible but rarely used in basic applications.

When the ratio is negative, the general term formula still works, but be careful with fractional exponents if you are finding a term between two given ones — that requires a real root, which may not exist for negative bases.

Frequently Asked Questions

How do I find the common ratio if I only have two non-consecutive terms?

Use the formula r = (a_m / a_k)^{1/(m−k)}. For example, if the 3rd term is 18 and the 6th term is 486, then r = (486/18)^{1/(6−3)} = 27^1/3 = 3. You need to know the positions (which term number each value is).

What if the common ratio is 0?

A geometric sequence with a common ratio of 0 would look like 5, 0, 0, 0, … — but this is degenerate because after the first term, every term is zero, and the ratio from the second to third term is 0/0, which is undefined. In practice, a geometric sequence is defined with a non-zero common ratio.

Can the common ratio be a fraction?

Yes. For example, the sequence 8, 4, 2, 1 has a common ratio of 1/2 (0.5). The calculator accepts decimal inputs, so you can enter 8 and 4 and get 0.5.

How is the common ratio different from the common difference?

The common ratio is used in geometric sequences and is found by dividing consecutive terms. The common difference is used in arithmetic sequences and is found by subtracting consecutive terms. For example, in 3, 6, 12, the common ratio is 2; in 3, 6, 9, the common difference is 3.

What does a negative common ratio mean?

A negative ratio makes the terms alternate in sign. For instance, with a ratio of −2 and first term 1, the sequence is 1, −2, 4, −8, … The absolute value tells you the growth factor, but the sign flips each time.

Can I use this calculator for a sequence that starts with a negative number?

Yes, the calculator works with negative terms as long as you don’t divide by zero. For example, first term −4, second term 12 gives a ratio of 12 ÷ (−4) = −3.

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in a | First term | - | 2 | 0
in b | Second term | - | 6 | 0
out r | Common ratio | - | b / a | 4
out next | Next term | - | b * (b / a) | 4
out formula | General term | - | a * (b/a)^(n-1) | 4