Introduction to Sequences

BASIC DEFINITION AND CONCEPT

A sequence is an ordered list of numbers where each element is determined by a specific place or position. We usually denote sequences with the symbol aₙ, where n ∈ ℕ (natural numbers) represents the sequential number (index), and aₙ is the n-th term of the sequence. A sequence can be defined in two ways:

1. With an explicit formula, where each term is expressed directly as a function of n.
2. Recursively, where the first (or first few) terms are defined, along with a rule for calculating the next term from the preceding term(s).

Example of a sequence defined by an explicit formula:
aₙ = n² gives the sequence (1, 4, 9, 16, 25, …)

TYPES OF SEQUENCES

ARITHMETIC SEQUENCE

Each subsequent term differs from the previous one by the same constant number 'd' (common difference):
aₙ = a₁ + (n – 1) * d
Example: 3, 7, 11, 15, … (where d = 4)

GEOMETRIC SEQUENCE

Each subsequent term is obtained by multiplying the previous term by the same constant number 'q' (common ratio):
aₙ = a₁ * q^(n–1)
Example: 2, 4, 8, 16, … (where q = 2)

CONSTANT SEQUENCE

All terms are equal: aₙ = c (where c is a constant)

ALTERNATING SEQUENCE

The values of the terms alternate in a prescribed pattern – often in sign.
Example: aₙ = (–1)ⁿ gives the sequence (–1, 1, –1, 1, …)

WAYS OF DEFINING SEQUENCES

• EXPLICITLY:
  We provide a formula for the general term aₙ as a function of n.
  Example: aₙ = 3n – 1
• RECURSIVELY:
  We define the initial term (or several initial terms) and a rule on how to obtain the next term.
  Example: a₁ = 2, aₙ₊₁ = aₙ + 5

INCREASING, DECREASING, AND BOUNDEDNESS

A sequence is:

• INCREASING if aₙ < aₙ₊₁ for all n.
• DECREASING if aₙ > aₙ₊₁ for all n.
• BOUNDED if there exist real numbers m and M such that: m ≤ aₙ ≤ M for all n.

Example of an increasing sequence: aₙ = n
Example of a bounded sequence: aₙ = 1/n (bounded between 0 (exclusive for n>0) and 1 (inclusive))

CONCLUSION

Sequences represent a fundamental concept in mathematical analysis and discrete mathematics. They allow for a structured notation of numerical series, the study of their properties and behavior, and preparation for further concepts such as series, limits of sequences, and functional analysis. Understanding different forms of sequences and ways of defining them is key for further work in many mathematical fields.