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Common Ratio

Common Ratio

For a geometric sequence or geometric series, the common ratio is the ratio of a term to the previous term. This ratio is usually indicated by the variable r.

 

Example: The geometric series 3, 6, 12, 24, 48, . . . has common ratio r = 2.

 

See also

Infinite geometric series

Key Formula

r=anan−1r = \frac{a_{n}}{a_{n-1}}
Where:
  • rr = The common ratio
  • ana_n = Any term in the geometric sequence
  • an−1a_{n-1} = The term immediately before a_n

Worked Example

Problem: Find the common ratio of the geometric sequence 5, 20, 80, 320, ...
Step 1: Pick any consecutive pair of terms. Divide the second term by the first term.
r=205=4r = \frac{20}{5} = 4
Step 2: Verify with another pair to confirm the ratio is constant.
8020=4and32080=4\frac{80}{20} = 4 \quad \text{and} \quad \frac{320}{80} = 4
Step 3: Since every consecutive ratio equals 4, the sequence is geometric with a common ratio of 4.
Answer: The common ratio is r = 4.

Another Example

Problem: Find the common ratio of the geometric sequence 81, 27, 9, 3, ...
Step 1: Divide the second term by the first term.
r=2781=13r = \frac{27}{81} = \frac{1}{3}
Step 2: Check with the next pair of terms.
927=13\frac{9}{27} = \frac{1}{3}
Step 3: The ratio is consistent. Note that the common ratio can be a fraction between 0 and 1, which causes the terms to decrease.
Answer: The common ratio is r = 1/3.

Frequently Asked Questions

Can the common ratio be negative?
Yes. When the common ratio is negative, the terms alternate between positive and negative values. For example, the sequence 2, −6, 18, −54, ... has a common ratio of r = −3. Each term flips sign because you multiply by a negative number.
What happens when the common ratio is between −1 and 1?
When |r| < 1, the terms get closer and closer to zero. This is important because it is the condition under which an infinite geometric series converges to a finite sum. For instance, with r = 1/2, the terms shrink: 8, 4, 2, 1, 1/2, ...

Common Ratio vs. Common Difference

The common ratio is the constant multiplier between consecutive terms of a geometric sequence (e.g., 3, 6, 12 has r = 2). The common difference is the constant amount added between consecutive terms of an arithmetic sequence (e.g., 3, 7, 11 has d = 4). Geometric sequences use multiplication; arithmetic sequences use addition.

Why It Matters

The common ratio determines the entire behavior of a geometric sequence. When |r| > 1, terms grow without bound (exponential growth); when |r| < 1, terms shrink toward zero (exponential decay). These patterns model real-world phenomena like compound interest, population growth, radioactive decay, and the convergence of infinite geometric series.

Common Mistakes

Mistake: Dividing a term by the next term instead of dividing the next term by the previous term.
Correction: Always divide a term by the one before it: r = a_n / a_{n−1}. Reversing the order gives you 1/r instead of r.
Mistake: Confusing common ratio with common difference and subtracting consecutive terms instead of dividing.
Correction: Subtraction finds the common difference of an arithmetic sequence. For a geometric sequence, you must divide consecutive terms to find the common ratio.

Related Terms