Understanding Notations for Sequences

By Mark Zegarelli

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:

Three mathematical sequences.

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

Referencing specific terms in a sequence by using the subscript.

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

  • The notation {an} with braces refers to the entire sequence.

  • The notation an without braces refers to the nth 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:

Three sequences defined with a rule based on n

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

Mathematical sequences including the first few terms plus a rule for finding the nth term

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

Sequences with appending starting and ending values.

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.