Lesson 19.6

Finite Geometric Series

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

Introduction

A finite geometric series is the sum of a fixed number of terms from a geometric sequence. Unlike arithmetic series, the sum formula involves multiplying by powers of the ratio—giving exponential behavior that's crucial for calculating compound interest and loan payments.

Prerequisite Connection

You understand geometric sequences and can find .

Today's Increment

We derive and apply .

Why This Matters

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 ≈