Geometric Series: Applications and Real-World Problems

Lesson 2 of 2

In this lesson:

• Apply the formula to series with , , and
• Identify parameters from real-world context using a timeline
• Solve savings and geometric growth problems

What You Will Learn in This Lesson

By the end of this lesson, you should be able to:

1. Apply the formula for all three cases: , ,
2. Choose the right formula form
3. Identify , , from real-world context
4. Verify by direct addition for small

Back to the Savings Problem

In Lesson 1 we opened with: $200/month for 24 months at monthly interest.

We asked: "How many calculations would you need?" — answer: 24.

Today we solve it. But first, let's make sure the formula is ready.

Both Forms of the Formula

• : first form — numerator and denominator both positive
• : second form — both positive, avoids sign errors
• :

Worked Example: Series with

, , : Sum of

Direct check: ✓

Worked Example: Series with

, , : Sum of

Direct check: ✓

Worked Example: The Special Case

, , : Sum of

Attempt formula: — undefined

Correct:

Always check before substituting.

Quick Check: Apply the Formula

Find the sum: .

Identify , , . Choose the right form. Compute .

Work it out before the next slide.

Practice: Six Series — Identify and Compute

Identify , , , then compute :

5. (10 terms)

Practice: Six Series — Check Your Answers

1. , , :
2. , , :
3. , , :
4. , , :
5. , , :
6. , , :

A Three-Step Method for Real-World Series

1. Draw a timeline — place each payment or event in order
2. Write the first three terms algebraically — find and
3. Count the terms — confirm

Why the First Term Is Not the Payment

$200/month for 24 months at monthly interest:

• Last deposit: earns 1 month →
• Next-to-last: earns 2 months →
• First deposit: earns 24 months →

First term: , not .

Savings Plan: Timeline and Setup

$200/month for 24 months at monthly interest.

Savings Plan: Applying the Formula

, ,

Verify with :

Formula: ✓

Find the Error in This Solution

Student's work: $200/month for 24 months at monthly interest.

", , "

What is wrong? How large is the error?

Worked Example: The Bouncing Ball

Dropped from 10 m, each bounce of previous. , , .

(Infinite sum would be m — 8 bounces reach of it.)

Practice: Ball and Savings Plan Problems

1. Ball dropped from 20 m, bouncing to height each time. Total downward distance over 6 bounces?

2. $300/month for 12 months at monthly interest. Total after 12 months?

Draw a timeline for problem 2. Identify , , first.

Practice: Ball and Savings Plan Answers

Problem 1: , ,

Problem 2: , ,

Full Procedure: No Prompts This Time

A ball dropped from 20 m, bouncing to height, for 6 bounces.

1. Write first three terms algebraically
2. Identify , ,
3. Apply the correct formula form
4. Verify with

What You Can Now Do: Closing Thoughts

Parameter identification is the skill; the formula is the tool.

• Draw a timeline first — it makes , , visible
• In savings annuities, , not the bare payment
• When : infinite sum converges to