Sequences

A sequence is a function whose domain is the set of positive integers. Simply put: a list of numbers arranged according to a rule.

Each element is generally denoted by the expression \(a_n\), where n indicates the order number.

Arithmetic Sequence

An arithmetic sequence is one in which the difference between any two consecutive terms is constant. This difference is denoted by d.

For example, if the first term is 2 and the difference is 3, the sequence looks like this: 2, 5, 8, 11, 14, ...

This is the sum of the first n terms of the arithmetic sequence.

Geometric Sequence

A geometric sequence is one in which the ratio of any two consecutive terms is constant. This ratio is denoted by q.

For example, if the first term is 3 and the ratio is 2, the sequence looks like: 3, 6, 12, 24, 48, ...

This is the sum of the first n terms of the geometric sequence.

Types of Sequences

• Increasing sequence: each subsequent term is larger than the previous.
• Decreasing sequence: each subsequent term is smaller than the previous.
• Monotonic sequence: always increasing or always decreasing.
• Bounded sequence: has an upper and/or lower bound.

Practical Examples

• In finance: monthly savings can form an arithmetic sequence.
• Mortgage repayments: often follow the logic of a geometric sequence.
• In physics: for certain motions, the distance traveled forms a sequence.

Practice Exercise