Common Ratio in Geometric Progression- Definition and Examples

What Is a Common Ratio in Geometric Progression?

A common ratio is the fixed number you multiply by to get each term in a geometric sequence. Every term after the first is found by multiplying the previous term by this constant value.

That's it. That's the whole concept.

The word "common" here just means "shared" or "the same across all terms." The ratio stays constant throughout the entire sequence, which is what makes it geometric.

The Formula

If your first term is a and your common ratio is r, then:

First term: a
Second term: ar
Third term: ar²
Fourth term: ar³
nth term: arⁿ⁻¹

See the pattern? The exponent on r is always one less than the term number.

Finding the Common Ratio

To find r, divide any term by the term right before it:

r = a₂ ÷ a₁

Or equivalently:

r = aₙ ÷ aₙ₋₁

You can use any two consecutive terms. If the sequence is truly geometric, you'll get the same answer every time.

Examples

Example 1: Positive Common Ratio

Sequence: 3, 6, 12, 24, 48, ...

6 ÷ 3 = 2

12 ÷ 6 = 2

24 ÷ 12 = 2

The common ratio is r = 2. Each term doubles the previous one.

Example 2: Fractional Common Ratio

Sequence: 100, 50, 25, 12.5, ...

50 ÷ 100 = 0.5

25 ÷ 50 = 0.5

The common ratio is r = 0.5. Each term is half the previous one.

Example 3: Negative Common Ratio

Sequence: 5, -15, 45, -135, ...

-15 ÷ 5 = -3

45 ÷ -15 = -3

The common ratio is r = -3. The terms alternate between positive and negative, and their absolute values grow.

Example 4: Common Ratio Between 0 and 1

Sequence: 1000, 100, 10, 1, ...

100 ÷ 1000 = 0.1

10 ÷ 100 = 0.1

The common ratio is r = 0.1. The terms shrink toward zero.

What the Common Ratio Does to Your Sequence

The value of r determines how the sequence behaves:

Value of r	Behavior	Example
\|r\| > 1	Terms grow without bound	2, 4, 8, 16, ... (r = 2)
r = 1	All terms are identical	5, 5, 5, 5, ... (r = 1)
0 < r < 1	Terms shrink toward zero	100, 50, 25, ... (r = 0.5)
r = 0	All terms after first are zero	7, 0, 0, 0, ... (r = 0)
-1 < r < 0	Terms shrink toward zero, alternating signs	10, -5, 2.5, ... (r = -0.5)
r = -1	Terms alternate between two values	6, -6, 6, -6, ... (r = -1)
\|r\| > 1	Terms grow in absolute value, alternating signs	3, -9, 27, -81, ... (r = -3)

How to Identify a Geometric Sequence

Not every sequence is geometric. Here's how to check:

Pick any two consecutive terms
Divide the second by the first
Pick another pair of consecutive terms
Divide again
If both answers match, you have a geometric sequence with that common ratio

If the ratios don't match, it's not geometric. It might be arithmetic (where you add a constant difference instead).

Common Ratio vs Common Difference

Don't confuse these two:

Arithmetic sequence: You add the same number each time. That number is the common difference (d).
Geometric sequence: You multiply by the same number each time. That number is the common ratio (r).

Example of arithmetic: 5, 10, 15, 20, ... (add 5 each time, d = 5)

Example of geometric: 5, 10, 20, 40, ... (multiply by 2 each time, r = 2)

Quick Reference Table

Given	Find	Formula
a₁ and r	Any term (aₙ)	aₙ = a₁ × rⁿ⁻¹
Two consecutive terms	Common ratio (r)	r = a₂ ÷ a₁
First term and last term, r	Number of terms (n)	n = log(last/a₁) ÷ log(r) + 1
First and last term, n	Common ratio (r)	r = (last/a₁)^(1/(n-1))

Getting Started: Solve It Yourself

Try finding the common ratio for this sequence:

2, 8, 32, 128, ...

Step 1: Divide 8 by 2 → r = 4

Step 2: Check: 32 ÷ 8 = 4 ✓

Step 3: Check: 128 ÷ 32 = 4 ✓

Answer: r = 4

Now find the 7th term using aₙ = a₁ × rⁿ⁻¹:

a₇ = 2 × 4⁶ = 2 × 4096 = 8192

Where Common Ratios Appear

You see geometric sequences with common ratios in:

Finance: Compound interest grows by a common ratio each period
Biology: Cell division often doubles (r = 2)
Physics: Radioactive decay uses fractional ratios
Computer science: Binary search cuts the problem in half each step (r = 0.5)
Population growth: Often modeled with r > 1

The concept shows up everywhere once you know what to look for.