# Understanding Notations for Sequences

Understanding sequences is an important first step toward understanding series. The simplest notation for defining a sequence is a variable with the subscript *n* surrounded by braces. For example:

You can reference a specific term in the sequence by using the subscript:

Make sure you understand the difference between notation with and without braces:

The notation {

*a*}_{n}*with*braces refers to the*entire*sequence.The notation

*a*_{n}*without*braces refers to the*n*th term of the sequence.

When defining a sequence, instead of listing the first few terms, you can state a rule based on *n**.* (This is similar to how a function is typically defined.) For example:

Sometimes, for increased clarity, the notation includes the first few terms plus a rule for finding the *n*th term of the sequence. For example:

This notation can be made more concise by appending starting and ending values for *n**:*

This last example points out the fact that the initial value of *n* doesn’t have to be 1, which gives you greater flexibility to define a number series by using a rule.

Don’t let the fancy notation for number sequences get to you. When you’re faced with a new sequence that’s defined by a rule, jot down the first four or five numbers in that sequence. After you see the pattern, you’ll likely find that a problem is much easier.