Lesson 19.3

Arithmetic Sequences

Identifying patterns based on a common difference d.

Introduction

An arithmetic sequence is the simplest type of pattern: each term is obtained by adding the same constant to the previous term. This constant is called the common difference . The sequence grows (or shrinks) at a steady, linear rate—making it predictable and easy to work with.

Prerequisite Connection

You understand sequences, explicit formulas, and sigma notation.

Today's Increment

We identify arithmetic sequences and derive the explicit formula .

Why This Matters

Arithmetic sequences model linear growth: hourly wages, stair steps, evenly-spaced events, and depreciation schedules.

Key Concepts

Definition: Arithmetic Sequence

A sequence where each term differs from the previous by a constant :

Explicit Formula

: n-th term

: first term

: common difference

Finding the Common Difference

Subtract any term from the next:

For 2, 5, 8, 11, ...:

Worked Examples

Example 1: Finding the Formula (Basic)

Find the explicit formula for: 3, 7, 11, 15, ...

Identify and :

Apply the formula:

Example 2: Finding a Specific Term (Intermediate)

Find the 50th term of an arithmetic sequence where and

Use the formula with n = 50:

Simplify:

Example 3: Finding Terms Given Two Points (Advanced)

Find and given and

Find d using the relationship:

→ →

Find using :

→

Formula:

Common Pitfalls

Using instead of

The formula is , not . At , you should get .

Confusing arithmetic with geometric

Arithmetic: ADD the same value. Geometric: MULTIPLY by the same value. They're very different!

Sign errors with negative

If , the sequence decreases: 10, 8, 6, 4, ...

Real-World Application

Stadium Seating

Stadiums often have rows with increasing numbers of seats. If the first row has 20 seats and each subsequent row has 2 more seats, the n-th row has seats. Event planners use this to calculate total capacity and ticket revenue.

Row 1: 20 seats → Row 25: seats