WHAT ARE SEQUENCES?

Sequences are a basic concept in mathematics, representing a series of numbers or elements arranged in a specific order. Each element in a sequence is known as a term of the sequence.

DEFINITION AND CHARACTERISTICS

A sequence is a list of elements, such as numbers or functions, that follow a particular pattern or rule. Each term of the sequence is usually denoted by a_n (read as "a sub n"), where 'n' is a natural number representing the position of the term in the sequence.

TYPES OF SEQUENCES

ARITHMETIC SEQUENCE: The characteristic of this sequence is that the difference between consecutive terms is constant (this is called the common difference). Example: 2, 4, 6, 8, … (common difference is 2)
GEOMETRIC SEQUENCE: Here, the ratio between consecutive terms is constant (this is called the common ratio). Example: 3, 6, 12, 24, … (common ratio is 2)
HARMONIC SEQUENCE: Each term is the reciprocal of the terms in an arithmetic sequence. (More accurately, the reciprocals of the terms form an arithmetic sequence). Example: 1, 1/2, 1/3, 1/4, … (the reciprocals 1, 2, 3, 4,... form an arithmetic sequence)
FIBONACCI SEQUENCE: Each term is the sum of the two preceding terms. Example: 0, 1, 1, 2, 3, 5, 8, … (assuming it starts with 0 and 1)
ALTERNATING SEQUENCE: Terms alternate between positive and negative values (or other patterns). Example: -1, 2, -3, 4, -5, …
CONSTANT SEQUENCE: All terms have the same value. Example: 4, 4, 4, 4, …

PROPERTIES

Sequences can either converge or diverge.

A convergent sequence has a limit as 'n' tends towards infinity (meaning the terms get closer and closer to a specific finite value).
A divergent sequence either does not have a finite limit, or its limit tends towards infinity or negative infinity.

CONCLUSION

Sequences are crucial for understanding mathematical patterns and structures. They are used in various mathematical disciplines, from algebra to advanced analysis, and are a fundamental concept for the study of series and limits.