Lesson 19.5

Geometric Sequences

Identifying patterns based on a common ratio r.

Introduction

A geometric sequence grows by multiplication: each term is the previous term times a constantcommon ratio . This creates exponential growth (or decay), which models compound interest, population dynamics, and radioactive decay.

Prerequisite Connection

You understand arithmetic sequences (addition pattern).

Today's Increment

We identify geometric sequences and use .

Why This Matters

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: