In mathematics, sequences and series are fundamental concepts that play a significant role in various branches of the subject. Let's start by understanding the difference between them.

Sequences are ordered lists of numbers that follow a specific pattern. Each number in the sequence is called a term, and we can represent a sequence as (a1, a2, a3, ...), where a1, a2, a3, ... are the terms of the sequence.

Series, on the other hand, is the sum of the terms in a sequence. It represents the cumulative effect of adding all the terms together.

Now, let's explore two types of sequences: arithmetic and geometric sequences.

Arithmetic sequences have a constant difference between consecutive terms. For example, consider the sequence (2, 5, 8, 11, ...), where the common difference is 3. Each term can be obtained by adding 3 to the previous term.

Geometric sequences have a constant ratio between consecutive terms. For instance, consider the sequence (2, 6, 18, 54, ...), where the common ratio is 3. Each term can be obtained by multiplying the previous term by 3.

By understanding sequences and series, we can unlock various mathematical phenomena and apply them to solving real-world problems!