Lesson 19.6

Finite Geometric Series

Using the ratio-based sum formula for a closed set of terms.

Introduction

Using the ratio-based sum formula for a closed set of terms.

Past Knowledge

You understand geometric sequences and can find .

Today's Goal

We derive and apply .

Future Success

Geometric series calculate loan totals, investment growth, and probability distributions.

Key Concepts

Sum Formula (r ≠ 1)

Both forms are equivalent; use whichever avoids negatives.

Special Case: r = 1

If , every term is , so:

Derivation Trick

Multiply the sum by r, subtract from original, and solve:

→

Worked Examples

Example 1: Basic Sum (Basic)

Find the sum: 3 + 6 + 12 + 24 + 48

, ,

Example 2: Decay Series (Intermediate)

Find for 100 + 50 + 25 + ...

, ,

Example 3: Sigma Notation (Advanced)

Evaluate

, , terms (k = 0 to 7)

Common Pitfalls

Using r = 1 in the formula

The formula divides by (1-r). If r = 1, use instead.

Counting terms wrong

has 8 terms (0,1,2,...,7), not 7!

Real-World Application

Investment Growth

Investing $1000 yearly at 8% interest: each year's deposit grows at different rates. The total is a geometric series!

10-year total ≈