Lesson 19.5

Geometric Sequences

Identifying patterns based on a common ratio r.

Introduction

Identifying patterns based on a common ratio r.

Past Knowledge

You understand arithmetic sequences (addition pattern).

Today's Goal

We identify geometric sequences and use .

Future Success

Geometric sequences model compound interest, bacterial growth, and any scenario with percentage-based change.

Key Concepts

Definition

A sequence where each term is the previous times a constant :

Explicit Formula

: n-th term

: first term

: common ratio

Finding the Common Ratio

For 3, 6, 12, 24:

Worked Examples

Example 1: Finding a Term (Basic)

Find the 8th term of 2, 6, 18, 54, ...

Example 2: Decay Sequence (Intermediate)

Find the 5th term of 100, 50, 25, ...

Example 3: Finding r and n (Advanced)

If and , find

→ →

Common Pitfalls

Using n instead of n-1 in exponent

Formula is , not .

Confusing with arithmetic

Geometric: MULTIPLY by r. Arithmetic: ADD d.

Real-World Application

Compound Interest

$1000 at 5% annual interest:

After 10 years: