A mathematical sequence is an ordered list of numbers that follow a specific pattern. The two most common types of sequences studied in high school and college algebra are arithmetic sequences and geometric sequences. Understanding the difference between them is key to solving progression problems.

What is an Arithmetic Sequence?

An arithmetic sequence is one in which the difference between consecutive terms is constant. This constant difference is called the common difference (d). You find any term by adding (or subtracting) the common difference to the previous term.

For example, the sequence 3, 7, 11, 15, 19... is arithmetic because the common difference (d) is +4. The formula for the nth term is: a_n = a_1 + (n - 1) * d.

What is a Geometric Sequence?

A geometric sequence is one in which each term is found by multiplying the previous term by a constant non-zero number. This constant multiplier is called the common ratio (r).

For example, the sequence 2, 6, 18, 54... is geometric because the common ratio (r) is 3. The formula for the nth term is: a_n = a_1 * r^(n - 1).