Geometric progression

Geometric sequences

Geometric progression sum visual proof

Geometric progression (also known as a geometric sequence) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 4, 8, 16, 32 is a geometric progression with a common ratio of 2. Geometric progressions are a fundamental concept in mathematics and have various applications in fields such as finance, physics, and computer science.

Definition

A geometric progression can be defined by two elements: the first term \(a\) and the common ratio \(r\), where \(r \neq 0\). The \(n\)th term of a geometric progression can be expressed as: \[a_n = a \cdot r^{(n-1)}\] where

• \(a_n\) is the \(n\)th term of the sequence,
• \(a\) is the first term of the sequence,
• \(r\) is the common ratio, and
• \(n\) is the term number.

Sum of a Geometric Progression

The sum of the first \(n\) terms of a geometric progression can be calculated using the formula: \[S_n = \frac{a(1 - r^n)}{1 - r}\] for \(r \neq 1\), and \[S_n = a \cdot n\] for \(r = 1\).

Properties

Geometric progressions have several important properties:

• If \(|r| < 1\), the terms of the progression get progressively closer to zero.
• If \(|r| > 1\), the terms of the progression increase in magnitude without bound.
• If \(r = -1\), the progression alternates between two values.
• The sum of an infinite geometric progression with \(|r| < 1\) converges to \(\frac{a}{1 - r}\).

Applications

Geometric progressions have wide-ranging applications across various disciplines:

• In finance, they are used to model exponential growth, such as compound interest.
• In physics, they describe phenomena with exponential decay or growth, such as radioactive decay.
• In computer science, algorithms with logarithmic complexity are often based on principles of geometric progressions.

Examples

1. A classic example of a geometric progression is the legend of the invention of chess, where the inventor asked for one grain of rice for the first square, two for the second, four for the third, and so on, doubling the amount for each of the 64 squares. 2. The binary system, fundamental to computer science, is based on a geometric progression with a common ratio of 2.

See Also

• Arithmetic progression
• Exponential function
• Logarithm
• Series (mathematics)