#businessarticle7 #mathematics #linearpatterns #geometricpatterns

Rules for linear and geometric patterns

By Lona Matshingana 

2026/01/01

Linear and geometric patterns are two fundamental sequences that differ in how they grow:

## Linear Patterns

A linear pattern adds (or subtracts) the same constant amount at each step. This constant is called the **common difference**, d.

**General form**: a, a+d, a+2d, a+3d, ...

Examples:

- 3, 7, 11, 15, 19, ... (common difference d = 4)

- 100, 95, 90, 85, ... (common difference d = -5)

The nth term follows the formula: **aₙ = a₁ + (n-1)d**

These patterns grow by adding, so they increase (or decrease) at a constant rate. Their graphs form straight lines, hence the name "linear."

## Geometric Patterns

A geometric pattern multiplies (or divides) by the same constant amount at each step. This constant is called the **common ratio**, r.

**General form**: a, ar, ar², ar³, ...

Examples:

- 2, 6, 18, 54, 162, ... (common ratio r = 3)

- 80, 40, 20, 10, 5, ... (common ratio r = 1/2)

The nth term follows the formula: **aₙ = a₁ · r^(n-1)**

These patterns grow by multiplying, so they can increase or decrease exponentially. With r > 1, they grow rapidly; with 0 < r < 1, they decay toward zero.

## Key Differences

Linear patterns have **constant absolute change** between terms, while geometric patterns have **constant proportional change**. Linear growth is steady and predictable; geometric growth accelerates (or decelerates) dramatically. This makes geometric patterns particularly important in modeling population growth, compound interest, radioactive decay, and many natural phenomena.

Thank you for reading!!!