Geometric Sequence
A geometric sequence (or geometric progression) is an ordered list 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. The nth term is given by aₙ = a₁ × r^(n−1). Geometric sequences model exponential growth and decay, compound interest, population doubling, and signal attenuation.
Key Formula
a_n = a_1 × r^(n-1)
LaTeX: a_n = a_1 \cdot r^{n-1}
| Symbol | Meaning | Unit |
|---|---|---|
| aₙ | nth term of the sequence | dimensionless |
| a₁ | First term | dimensionless |
| r | Common ratio (aₙ / aₙ₋₁) | dimensionless |
| n | Term number (positive integer) | dimensionless |
Worked Example
Problem
A geometric sequence has first term 3 and common ratio 4. Find the 6th term and the sum of the first 6 terms.
Solution
Step 1: 6th term: a₆ = 3 × 4^(6−1) = 3 × 4⁵ = 3 × 1024 = 3072. Step 2: Sum: S₆ = a₁(rⁿ − 1)/(r − 1) = 3(4⁶ − 1)/(4 − 1) = 3(4096 − 1)/3 = 4095.
Answer
a₆ = 3072; S₆ = 4095.
Examples of Geometric Sequences
| Sequence | First Term (a₁) | Common Ratio (r) | Behaviour |
|---|---|---|---|
| 2, 6, 18, 54, … | 2 | 3 | Exponential growth |
| 100, 50, 25, 12.5, … | 100 | 0.5 | Exponential decay |
| 1, −2, 4, −8, … | 1 | −2 | Alternating signs |
| 5, 5, 5, 5, … | 5 | 1 | Constant (r = 1) |
| 81, 27, 9, 3, 1, … | 81 | 1/3 | Decreasing fraction |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
Arithmetic Sequence
An arithmetic sequence (also called an arithmetic progression) is an ordered list of numbers in which the difference between consecutive terms is constant, called the common difference d. The nth term is given by aₙ = a₁ + (n−1)d, where a₁ is the first term. Arithmetic sequences model uniform increments such as salary increases, regular savings, and equally spaced physical measurements.
Exponential Function
An exponential function is a mathematical function of the form f(x) = aˣ, where the variable x appears as the exponent and the base a is a positive constant not equal to 1. These functions exhibit rapid growth (when a > 1) or decay (when 0 < a < 1) and are fundamental models for population growth, compound interest, radioactive decay, and many natural phenomena.
Mathematical Series
A mathematical series is the sum of the terms of a sequence, either finite or infinite. For a finite series the sum is always a specific number, while an infinite series converges to a finite limit only if its terms decrease fast enough. Series are foundational in calculus, number theory, physics, and engineering — from computing π to modelling oscillations and signal processing.
From Greek "geometrikos" (relating to geometry), as the sequence arises in geometric figures such as similar triangles whose sides grow by a fixed ratio. The Latin "progressio" (forward movement) completes the term.