Summary: Sequences and Their Notations

Key Equations

Formula for a factorial [latex]\begin{array}{l}0!=1\\ 1!=1\\ n!=n\left(n - 1\right)\left(n - 2\right)\cdots \left(2\right)\left(1\right)\text{, for }n\ge 2\end{array}[/latex]

Key Concepts

  • A sequence is a list of numbers, called terms, written in a specific order.
  • Explicit formulas define each term of a sequence using the position of the term.
  • An explicit formula for the [latex]n\text{th}[/latex] term of a sequence can be written by analyzing the pattern of several terms.
  • Recursive formulas define each term of a sequence using previous terms.
  • Recursive formulas must state the initial term, or terms, of a sequence.
  • A set of terms can be written by using a recursive formula.
  • A factorial is a mathematical operation that can be defined recursively.
  • The factorial of [latex]n[/latex] is the product of all integers from 1 to [latex]n[/latex]

Glossary

explicit formula a formula that defines each term of a sequence in terms of its position in the sequence

finite sequence a function whose domain consists of a finite subset of the positive integers [latex]\left\{1,2,\dots n\right\}[/latex] for some positive integer [latex]n[/latex]

infinite sequence a function whose domain is the set of positive integers

n factorial the product of all the positive integers from 1 to [latex]n[/latex]

nth term of a sequence a formula for the general term of a sequence

recursive formula a formula that defines each term of a sequence using previous term(s)

sequence a function whose domain is a subset of the positive integers

term a number in a sequence