1. Definition and Notation
What is a Geometric Progression?
A geometric progression (GP), also called a geometric sequence, is a sequence of non-zero numbers in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero constant called the common ratio, denoted $q$ (or sometimes $r$).
Formal Definition: A sequence $(b_1, b_2, b_3, \ldots)$ is a geometric progression if there exists a constant $q \neq 0$ such that
$$b_{n+1} = b_n \cdot q \quad \text{for all } n \geq 1, \quad b_1 \neq 0.$$
The sequence is typically written as:
$$b_1,\; b_1 q,\; b_1 q^2,\; b_1 q^3,\; \ldots$$Recursive vs. Explicit Definition
Recursive definition:
Defines each term in relation to the previous term: $$b_1 = a, \quad b_{n+1} = b_n \cdot q$$
Defines each term in relation to the previous term: $$b_1 = a, \quad b_{n+1} = b_n \cdot q$$
Explicit (closed-form) definition:
Defines each term directly in terms of its position $n$: $$b_n = b_1 \cdot q^{n-1}$$
Defines each term directly in terms of its position $n$: $$b_n = b_1 \cdot q^{n-1}$$
Behavior Based on the Common Ratio $q$
| Value of $q$ | Behavior | Example ($b_1 = 2$) |
|---|---|---|
| $q > 1$ | Terms grow without bound (exponential growth) | $2, 6, 18, 54, \ldots$ ($q=3$) |
| $q = 1$ | All terms are equal (constant sequence) | $2, 2, 2, 2, \ldots$ |
| $0 < q < 1$ | Terms decay toward zero | $2, 1, 0.5, 0.25, \ldots$ ($q=0.5$) |
| $-1 < q < 0$ | Terms alternate in sign and decay toward zero | $2, -1, 0.5, -0.25, \ldots$ ($q=-0.5$) |
| $q < -1$ | Terms alternate in sign and grow in absolute value | $2, -4, 8, -16, \ldots$ ($q=-2$) |