Series Success Slides Series Success
Building Wealth with Geometric Sums
Pattern Check
Before we add, let's identify.
Analyze this sequence:
5, 10, 20, 40, ...
What is the first term (\(a_1\))?
What is the common ratio (\(r\))?
1
A Geometric Sequence changes by multiplying the same number (\(r\)) every time.
2
A Series is what happens when we try to ADD those terms together.
Can you find the sum of the first 20 terms... instantly?
3 + 6 + 12 + 24 + ...
Stopping at \(n = 20\)
Adding manually would take forever. There has to be a better way to manage our "wealth."
Pattern Hunter
Let's look at the sequence: 1, 2, 4, 8, 16... (\(r = 2\))
Terms (\(n\)) The Series Sum (\(S_n\)) What is \(r^n - 1\)? 3 1 + 2 + 4 7 \(2^3 - 1 = 7\) 4 1 + 2 + 4 + 8 15 \(2^4 - 1 = 15\) 5 1 + 2 + 4 + 8 + 16 31 \(2^5 - 1 = 31\)
Wait! The sum is related to the ratio raised to the power of \(n\).
The Geometric Vault Key
\[S_n = \frac{a_1(1 - r^n)}{1 - r}\]
\(a_1\)
The Starting Value
\(r\)
Common Ratio
\(n\)
Number of Terms
Wealth Building: Saving
You invest $100 at the start of every year. Your account grows by 5% each year (\(r = 1.05\)).
How much do you have after 10 years?
Note: Finance formulas often look scary, but they are just geometric series!
Calculations
\(a_1\) 100
\(r\) 1.05
\(n\) 10
Equation Setup:
\(S_{10} = \frac{100(1 - 1.05^{10})}{1 - 1.05}\)
Investor Challenge
If a viral video starts with 1,000 views on Day 1 and triples every day, how many total views will it have after 7 days?
Identify
\(a_1, r, n\)
Solve
Use Formula
Series Success Worksheet Series Success
Geometric Series & Financial Growth
Name:
Date:
1. Pattern Hunter: The Power of Sums
Use your calculator to find the sums of the sequence 1, 2, 4, 8, 16... (\(r = 2\)). Then, compare the sum (\(S_n\)) to the value of \(2^n - 1\).
Terms (\(n\)) The Addition Problem Sum (\(S_n\)) Calculate: \(2^n - 1\) 2 terms 1 + 2 3 \(2^2 - 1 = 3\) 3 terms 1 + 2 + 4 7 \(2^3 - 1 = 7\) 4 terms ____________________ ________ ________________ 5 terms ____________________ ________ ________________
Reflect:
What do you notice about the relationship between the Sum (\(S_n\)) and the value of \(r^n - 1\)?
2. The Formula Vault
Geometric Sum Formula
\[S_n = \frac{a_1(1 - r^n)}{1 - r}\]
\(a_1\)
First Term
\(r\)
Common Ratio
\(n\)
# of Terms
3. Guided Growth
Example 1: The Daily Double
Find the sum of the first 10 terms of a series where the first term is 5 and the common ratio is 2.
a₁ =
r =
n =
Calculations / Final Answer
Example 2: Saving for Success
Imagine you save $200 at the start of every year. Your money earns 6% interest (\(r = 1.06\)). How much total money is in the account after 8 years?
Tip: Set up the formula: \(S_8 = \frac{200(1 - 1.06^8)}{1 - 1.06}\)
Show your work here
Example 3: Viral Reach
A video gets 1,000 views on its first day. Each day, the number of new views is 1.5 times the previous day (\(r = 1.5\)). What is the total number of views after 5 days?
Identify Values
Calculation
Quick Check
When would it be easier to use the formula instead of adding up each term individually?
Series Success Teacher Guide Teacher Facilitation Guide
Lesson: Series Success (Geometric Series Intervention)
Target Standard
CO HS.A-SSE.B.4
Objective
Students will derive and apply the formula for the sum of a finite geometric series to solve problems, specifically focusing on financial models (compound interest/savings) and exponential growth scenarios.
Lesson Flow
01
Pattern Check (5 min)
Review geometric sequences. Ensure students can identify \(a_1\) and \(r\).
02
The Sum Discovery (10 min)
Guided calculator exploration. Use the table to help students "see" the relationship between \(r^n\) and the total sum.
03
Formula Application (15 min)
Connect the abstract formula to financial savings. Emphasize that \(r\) in finance is usually \(1 + \text{rate}\).
Scaffolding Tips
Visual Aids: Use color-coding for \(a_1\), \(r\), and \(n\) in the formula and word problems.Calculator Mechanics: Model how to enter the entire numerator in parentheses before dividing.Misconception: Students often forget the difference between a sequence (list) and a series (sum). Remind them: "Series = Sum".
Small Group Progress Monitoring
Student Name Identifies \(a_1\), \(r\), \(n\) Sets up Formula Calculator Accuracy Finance Context
Key Questions to Ask
• "Why is the common ratio in a savings account usually 1.0something?"
• "What happens to the total sum if the common ratio gets larger?"
Observation Legend
Not Yet Emerging Mastered
