Geometric Progression: Complete Guide
1. Definition of a Geometric Progression
Definition: A geometric progression (GP) is a sequence of non-zero numbers where each term, starting from the second, equals the previous term multiplied by the same constant number. This constant is called the common ratio $q$ (from the English word "quotient"), and $q \neq 0$.
Mathematically, for every $n \geq 2$:
$$\frac{b_n}{b_{n-1}} = q, \quad \text{or equivalently} \quad b_n = b_{n-1} \cdot q$$
Examples of geometric progressions:
| Sequence | $b_1$ | $q$ |
|---|---|---|
| $2, 6, 18, 54, 162, \ldots$ | $2$ | $3$ |
| $100, 50, 25, 12.5, \ldots$ | $100$ | $0.5$ |
| $3, -6, 12, -24, 48, \ldots$ | $3$ | $-2$ |
| $1, 1, 1, 1, 1, \ldots$ | $1$ | $1$ |
2. Types of Geometric Progressions Based on $q$
When $q > 0$
All terms have the same sign as $b_1$.
- If $b_1 > 0$ and $q > 1$: the sequence is increasing
Example: $2, 6, 18, 54, \ldots$ ($q = 3$) - If $b_1 > 0$ and $0 < q < 1$: the sequence is decreasing
Example: $100, 50, 25, \ldots$ ($q = 0.5$) - If $q = 1$: the sequence is constant
Example: $5, 5, 5, 5, \ldots$
When $q < 0$
Terms alternate in sign (positive, negative, positive, …).
- Example: $3, -6, 12, -24, \ldots$ ($q = -2$)
- Example: $-1, 2, -4, 8, \ldots$ ($q = -2$)
Such sequences are neither purely increasing nor purely decreasing — they oscillate.
Interactive visualization — available in the full lesson