progressões

Progressões, in mathematics, refers to sequences of numbers where each term is related to the previous one by a specific rule. The most common types are arithmetic progressions and geometric progressions. An arithmetic progression is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference. For example, 2, 5, 8, 11, ... is an arithmetic progression with a common difference of 3. The general form of an arithmetic progression is a, a+d, a+2d, a+3d, ..., where 'a' is the first term and 'd' is the common difference.

A geometric progression, on the other hand, is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For instance, 3, 6, 12, 24, ... is a geometric progression with a common ratio of 2. 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.

These sequences have numerous applications in various fields, including finance, physics, and computer science. Formulas exist to calculate the nth term of both arithmetic and geometric progressions, as well as their sums. Understanding these fundamental concepts is crucial for further study in areas like calculus and discrete mathematics.