geometric progression

Thân thiện

A student draws a geometric progression on a chalkboard.

Definition

1. Noun:
   • A sequence 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. This creates a pattern of exponential growth or decay.

Usage

• The term is used primarily in mathematics to describe a specific type of numerical sequence.
• It is often contrasted with an , where a constant is added (not multiplied) to get the next term.
• It describes patterns in finance (compound interest), population growth, computer science, and physics.

Examples

• Noun:
  • The sequence 2, 6, 18, 54... is a geometric progression with a common ratio of 3.
  • The decay of a radioactive substance follows a geometric progression.
  • In the example "1-4-16-64-256- is the start of a geometric progression," the common ratio is 4.

Advanced Usage

• "in geometric progression": Used as an adverbial phrase to describe how something increases or decreases.
  • The bacteria multiplied in geometric progression, quickly overwhelming the host.

Variants and Related Words

• Geometric series (n): The sum of the terms of a geometric progression.
  • Calculating the sum of an infinite geometric series requires the ratio to be between -1 and 1.
• Exponential growth (n): A pattern of data that shows sharper increases over time, often modeled by a geometric progression.

Synonyms

• Geometric sequence: A direct synonym.
• Exponential sequence: A near synonym, emphasizing the rapid change characteristic of many geometric progressions.

Related Phrases

• Common ratio: The constant factor between consecutive terms of a geometric progression.
  • To find the next number, multiply by the common ratio.

Related Idioms

• "To grow geometrically": An idiom meaning to increase at an extremely rapid, exponential rate.
  • With the advent of the internet, the availability of information has grown geometrically.

Học thuật

Noun

1. (mathematics) a progression in which each term is multiplied by a constant in order to obtain the next term
   • 1-4-16-64-256- is the start of a geometric progression