An exponential function is a type of mathematical function in which the independent variable appears as an exponent. In other words, the output of the function is a result of repeatedly multiplying the input by a fixed number, called the base. The general form of an exponential function is y = ab^x, where a is a constant and b is the base.
A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number, called the common ratio. The general form of a geometric progression is a, ar, ar^2, ar^3, ..., where a is the first term and r is the common ratio. Geometric progressions are commonly used to model situations where the quantity grows or decays exponentially over time.
Exponential functions and geometric progressions are closely related, as they both involve repeated multiplication by a fixed number. In fact, a geometric progression can be expressed as an exponential function, and vice versa. For example, the geometric progression 2, 4, 8, 16 can be written as the exponential function y = 2^x.
Exponential functions and geometric progressions are used extensively in science to model natural phenomena, such as population growth, radioactive decay, and compound interest. They are also used in many areas of physics, engineering, and economics to describe relationships between variables that grow or decay exponentially.
The main properties of exponential functions and geometric progressions include a constant ratio between consecutive terms, a curved shape on a logarithmic scale, and an infinite domain and range. In addition, exponential functions have a unique point called the y-intercept, while geometric progressions have a finite sum if the common ratio is less than 1. These properties make them useful tools for analyzing and predicting various phenomena in science and mathematics.