1. Review of Geometric Progression
Definition
A geometric progression (GP) is a sequence $b_1, b_2, b_3, \ldots$ where each term is obtained by multiplying the previous term by a fixed nonzero number $q$ called the common ratio:
$$b_{n+1} = b_n \cdot q \quad \text{for all natural } n, \quad b_n \neq 0, \quad q \neq 0$$
Key Formulas
| Formula | Expression | Condition |
|---|---|---|
| $n$-th term | $$b_n = b_1 \cdot q^{n-1}$$ | $n \geq 1$ |
| Sum of first $n$ terms | $$S_n = \frac{b_1(1 - q^n)}{1 - q}$$ | $q \neq 1$ |
Examples
Example A: $1,\; 3,\; 9,\; 27,\; \ldots$ with $q = 3$
Terms grow without bound.
Terms grow without bound.
Example B: $1,\; \frac{1}{5},\; \frac{1}{25},\; \frac{1}{125},\; \ldots$ with $q = \frac{1}{5}$
Terms shrink toward zero.
Terms shrink toward zero.
2. Definition of Infinitely Decreasing Geometric Progression
Core Definition
A geometric progression is called infinitely decreasing if the absolute value of its common ratio is less than one:
$$|q| < 1$$
As $n$ increases, the terms $b_n = b_1 \cdot q^{n-1}$ approach zero because $|q|^{n-1} \to 0$.
$q = \frac{1}{2}$ ✓
$|q| = 0.5 < 1$
$|q| = 0.5 < 1$
$q = -\frac{1}{3}$ ✓
$|q| = \frac{1}{3} < 1$
$|q| = \frac{1}{3} < 1$
$q = 0.1$ ✓
$|q| = 0.1 < 1$
$|q| = 0.1 < 1$
Interactive: Watch how terms of a GP approach zero when $|q| < 1$
Interactive visualization — available in the full lesson
Drag the sliders to change $q$ and $b_1$. Notice how terms approach zero when $|q| < 1$.