Description: A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. This distinctive characteristic allows the terms of the sequence to grow or shrink exponentially, depending on the value of the ratio. For example, if we start with the number 2 and use a common ratio of 3, the resulting sequence would be 2, 6, 18, 54, and so on. Geometric progressions are fundamental in mathematics and have applications in various fields such as finance, biology, and physics. Their study allows for understanding phenomena involving exponential growth or decay, such as compound interest in finance or population growth in biology. Additionally, geometric progressions can be represented through mathematical formulas that facilitate the calculation of specific terms and the sum of their elements, making them valuable tools for solving complex problems. In summary, the geometric progression is a mathematical structure that reflects patterns of growth and decay, being essential for the analysis and modeling of various situations in the real world.
History: The concept of geometric progression dates back to antiquity, with records of its use in civilizations such as Babylonian and Greek. Greek mathematicians, such as Euclid, explored the properties of ratios and numerical sequences. However, it was during the Renaissance that the study of geometric progressions was formalized, thanks to the work of mathematicians like Fibonacci and his famous ‘Liber Abaci’ in 1202, which introduced concepts of numerical sequences to Europe. Over the centuries, interest in geometric progressions has grown, especially in the context of number theory and mathematical analysis.
Uses: Geometric progressions have multiple applications across various disciplines. In finance, they are used to calculate compound interest, where capital grows exponentially. In biology, they model population growth, where populations can multiply at a constant rate. They are also found in physics, in phenomena such as radioactive decay, where the amount of material decreases exponentially over time. Additionally, in computer science, they are used in algorithms and in complexity theory.
Examples: A classic example of a geometric progression is the calculation of compound interest. If an amount of money, say 1000 euros, is invested at an annual interest rate of 5%, the balance after one year would be 1000 * 1.05 = 1050 euros. After two years, it would be 1050 * 1.05 = 1102.50 euros, and so on. Another example can be found in biology, where a population of bacteria that doubles every hour can be modeled as a geometric progression, starting with a single bacterium and growing exponentially.
