What is a Geometric Progression?

In science and mathematics, we often encounter patterns where quantities grow or shrink by the same multiplying factor each time. This type of sequence is called a geometric progression.

Geometric Progression (GP)

A sequence of numbers where each term is found by multiplying the previous term by a fixed, constant value called the common ratio.

Unlike an arithmetic progression (where you add the same number each time), in a geometric progression you multiply by the same number each time.

Analogy

Imagine you start with 1 sheet of paper and fold it in half repeatedly. Each fold doubles the thickness — you multiply by 2 every time. After just 10 folds, the paper would be $2^{10} = 1024$ times its original thickness! That doubling pattern is a geometric progression in action. (In practice, it's almost impossible to fold a real sheet of paper more than 7 or 8 times — a great example of how mathematical models and physical reality can differ!)

Here are some everyday examples of geometric progressions:

A bacteria colony that doubles every hour: 1, 2, 4, 8, 16, ...
Radioactive decay, where half the atoms decay each year: 1000, 500, 250, 125, ...
Compound interest on a savings account
The spread of a viral post on social media

Note

Geometric progressions appear constantly in science — population growth, drug concentration in the body, earthquake intensity (the Richter scale), and sound intensity (decibels) all use geometric (multiplicative) thinking.

The Common Ratio

Common Ratio (r)

The fixed number by which each term in a geometric progression is multiplied to get the next term. It is found by dividing any term by the term before it: $r = \frac{term _{n + 1}}{term _{n}}$

The common ratio $r$ can be:

Greater than 1 → the sequence grows (e.g., population explosion)
Between 0 and 1 → the sequence shrinks (e.g., radioactive decay)
Negative → the sequence alternates between positive and negative values
Equal to 1 → every term is the same (not very interesting!)

Example

Finding the common ratio

Consider the sequence: 3, 6, 12, 24, 48, ...

Divide any term by the one before it:
$r = \frac{6}{3} = 2 or r = \frac{12}{6} = 2 or r = \frac{24}{12} = 2$

The common ratio is $r = 2$ . Each term is doubled.

Now try: 80, 40, 20, 10, 5, ...
$r = \frac{40}{80} = 0.5$
The common ratio is $r = 0.5$ . Each term is halved.

Warning

Always check the ratio using at least two consecutive pairs of terms to confirm it is truly geometric (the ratio must be the same throughout). If the ratio changes, it is not a geometric progression.

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