Arithmetic Sequences

Definition

An arithmetic sequence is a sequence where the difference between any two successive terms is a constant, called the common difference ($d$).

Why It Matters

They are the simplest models for constant growth and are essential for financial planning and physics. Recognizing these patterns allows for the accurate prediction of future states in linear systems.

Core Concepts

• General Term Formula: $a_n = a_1 + (n-1)d$

  • How to read: “The n-th term a n equals the first term a one, plus the quantity n minus one times the common difference d.”
  • Meaning: The $n$th term of a sequence that grows by adding $d$ each step—like a discrete linear function.

• Arithmetic Series ($S_n$): The sum of the first $n$ terms: $S_n = \frac{n}{2} (a_1 + a_n) \quad \text{or} \quad S_n = \frac{n}{2} [2a_1 + (n-1)d]$

  • How to read: “The sum of the first n terms S n equals n divided by two, times the sum of the first term a one and the n-th term a n. Alternatively, S n equals n over two, times the sum of twice a one and the quantity n minus one times d.”
  • Meaning: First form uses first and last term; second uses first term and common difference. Both give the total of $n$ evenly spaced terms.

Connected Concepts

• Geometric Sequence and Geometric Series
• Linear Functions
• Infinite Sequences
• Mathematical Induction
• First-Order Linear Differential Equations
• Linear Equations, Problem Solving in Mathematics
• Taylor Series
• Maclaurin Series
• Scale