Definition of a Geometric Progression
A geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number. This constant multiplier is called the common ratio (r).
Key Components and General Form
The fundamental elements of a geometric progression are its first term (denoted as 'a' or a₁) and its common ratio (r). The general form of a geometric progression can be expressed as a, ar, ar², ar³, ..., ar^(n-1), where 'n' represents the position of the term in the sequence.
A Practical Example
Consider the sequence: 3, 6, 12, 24, 48, ... Here, the first term (a) is 3. To find the common ratio (r), divide any term by its preceding term; for instance, 6 ÷ 3 = 2, 12 ÷ 6 = 2, and so on. In this example, the common ratio (r) is 2.
Importance and Applications
Geometric progressions are vital in various real-world applications. They are used to model phenomena involving exponential growth or decay, such as compound interest calculations in finance, the growth of populations, or the decrease in radioactive material over time. They also form the basis for understanding geometric series.