Geometric Sequence

Definition

A Geometric Sequence is a sequence 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 ($r$).

Why It Matters

Geometric sequences are the mathematical engine of compound growth and decay. Whether mapping the trajectory of an epidemic, the half-life of a radioactive isotope, or the scaling of a network, understanding the sequence of values at discrete steps is essential.

Core Concepts

• The $n$-th Term Formula: The general term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$
  • How to read: “The term a n is equal to a one times r raised to the quantity n minus one.”
  • Meaning: Each term multiplies the first term by $r$ raised to the $(n-1)$th power; $r = \frac{a_n}{a_{n-1}}$ is the constant ratio.

Connected Concepts

• Geometric Series
• Exponential Functions
• Arithmetic Sequences