Sequence Definition

Definition

A sequence is a function whose domain is the set of positive integers ($\mathbb{Z}^+$) and whose range is a subset of the real numbers. It is an ordered list of numbers where each number is called a term.

• How to read: “The domain is the positive integers Z plus.”
• Meaning: A sequence assigns one real number to each counting number $n = 1, 2, 3, \dots$

Why It Matters

Sequences are the ‘building blocks’ of discrete mathematics; they allow us to transition from continuous models to indexed, step-by-step progressions, providing the foundation for series and algorithmic loops.

Core Concepts

• Terms: The individual elements, denoted as $a_1, a_2, a_3, \dots, a_n$.
• Summation Notation ($\Sigma$): A compact way to represent the sum of terms: $\sum_{k=1}^{n} a_k = a_1 + a_2 + \dots + a_n$
  • How to read: “The sum from k equals one to n of a k is equal to a one plus a two, and so on, up to a n.”
  • Meaning: Sigma adds every term in the sequence from index 1 through $n$.
• Fundamental Sum formulas:
  • $\sum_{k=1}^{n} c = cn$ - How to read: “Sum of c from k=1 to n equals c times n.” - Meaning: Adding a constant $n$ times.
  • $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ - How to read: “Sum of k equals n times (n+1) over two.” - Meaning / when to use: Sum of first $n$ integers—Gauss’s formula.

Connected Concepts

• Sequence Recursion
• Arithmetic Sequences