Sequences and Series

A sequence is a list of numbers (or other objects) that typically follow a specific pattern. Each number in a sequence is called a term.

A series is the sum of the terms of a sequence. While a sequence lists the numbers, a series adds them together.

Sequences

A sequence can be finite or infinite. The terms in a sequence can be generated by an explicit formula (like an = n^2 for the sequence of squares), or by a recursive formula (like an = an-1 + 2 for the sequence of even numbers).

Two important types of sequences are arithmetic and geometric sequences:

• Arithmetic sequences are sequences where each term is a certain number (the common difference) more than the previous term.

• Geometric sequences are sequences where each term is a certain factor (the common ratio) times the previous term.

Series

Just like sequences, series can be finite or infinite. Adding up the first n terms of a sequence gives a finite series, while adding up all the terms of a sequence gives an infinite series.

The sum of a finite arithmetic series can be found with the formula n/2 * (a1 + an), where n is the number of terms, a1 is the first term, and an is the last term.

The sum of a finite geometric series can be found with the formula a1 * (1 - r^n) / (1 - r), where a1 is the first term, r is the common ratio, and n is the number of terms.

For an infinite geometric series, the sum can be found (if it exists) with the formula a1 / (1 - r).

Applications

Sequences and series are used in many areas of mathematics and the physical sciences. They can be used to model population growth, calculate interest, solve differential equations, and more. They also form the basis for calculus, where infinite series are used to define integrals and derivatives.