Finite Series Slides FINITE SUMS
Modeling Geometric Growth
Pay It Forward
Imagine you do a favor for 3 people. Tomorrow, each of those 3 people does a favor for 3 more people. This continues for 10 days.
Day 0
1
Day 1
3
Day 2
9
"How many total favors have been done by Day 10?"
The Geometric Series
A finite geometric series is the sum of terms in a geometric sequence:
\[ S_n = a + ar + ar^2 + ar^3 + \dots + ar^{n-1} \]
a
First Term
r
Common Ratio
Deriving the Shortcut
1
Write the sum: \[ S_n = a + ar + ar^2 + \dots + ar^{n-1} \]
2
Multiply by \(r\): \[ rS_n = ar + ar^2 + ar^3 + \dots + ar^n \]
3
Subtract: \[ S_n - rS_n = a - ar^n \]
4
Factor and solve: \[ S_n(1-r) = a(1-r^n) \]
\[ S_n = \frac{a(1-r^n)}{1-r} \]
Quick Check
Problem 1
Find the sum of the first 8 terms of the series:
4, 12, 36, ...
Identify \(a\): ______
Identify \(r\): ______
Identify \(n\): ______
Back to the Favors
Initial favor: \(a = 1\)
Daily multiplier: \(r = 3\)
Duration: \(n = 11\) (Day 0 to 10)
\[ S_{11} = \frac{1(1-3^{11})}{1-3} \]
= 88,573 favors!
Series Spark Worksheet SERIES SPARK
Calculus: Finite Geometric Series
Name:
Date:
Reference Box
For a series with first term \(a\) and common ratio \(r\):
\[ S_n = \frac{a(1-r^n)}{1-r} \]
1
Computational Core
1. Evaluate the sum of the first 10 terms of the geometric series: \( 5 + 10 + 20 + \dots \)
\( a = \text{______} \)
\( r = \text{______} \)
\( n = \text{______} \)
Show Work
2. Find the sum: \[ \sum_{k=1}^{8} 12(0.5)^{k-1} \]
Show Work
2
Population Shift
A rare orchid population in a botanical garden is growing. Currently, there are 20 orchids. Each year, the population grows by 15%. To track the "total botanical effort" required over a decade, the garden director wants to know the sum of the orchid population counts across the next 12 years.
A. Write a geometric series expression representing the total orchid-years for the 12-year period.
B. Calculate the total sum. Round to the nearest orchid.
C. Critical Thinking: If the garden reached its capacity of 1,000 orchids in the 20th year, would the geometric series still be an accurate model? Why or why not?
3
The Reverse Engineering
The sum of the first \(n\) terms of a geometric series is 6,560. If \(a = 2\) and \(r = 3\), find the value of \(n\).
Puzzle Solver Section
Infinite Infinity Slides INFINITE LIMITS
When Infinity Has a Boundary
Zeno's Paradox
To walk across a room, you must first walk halfway. Then you must walk half of the remaining distance, and so on.
"Can you ever reach the wall if you have an infinite number of steps to take?"
1/2
1/2 + 1/4
1/2 + 1/4 + 1/8
... + (1/2)^n ...
The Condition for Convergence
CONVERGENT
|r| < 1
The terms shrink toward zero fast enough for the sum to be finite.
DIVERGENT
|r| ≥ 1
The terms grow or stay the same. The sum explodes to infinity.
The Magic Formula
As \(n\) goes to infinity, \(r^n\) goes to \(0\) (if \(|r| < 1\)).
\[ S_{\infty} = \frac{a}{1-r} \]
Example 1
\( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \)
\( a=1/2, r=1/2 \implies S = \frac{0.5}{1-0.5} = 1 \)
Example 2
\( 0.3333\dots = \frac{3}{10} + \frac{3}{100} + \dots \)
\( a=0.3, r=0.1 \implies S = \frac{0.3}{0.9} = \frac{1}{3} \)
Zeno Paradox Activity INFINITY BOUND
Exploration: Infinite Geometric Series
STUDENT NAME: ___________________________
Task 1: The Gateway
Determine if each series converges or diverges. If it converges, find the sum to infinity.
A) \( 8 + 6 + 4.5 + 3.375 + \dots \)
Ratio (r): _________
CONVERGE
DIVERGE
Sum (\(S_{\infty}\)):
B) \( 1 - 2 + 4 - 8 + \dots \)
Ratio (r): _________
CONVERGE
DIVERGE
Sum (\(S_{\infty}\)):
Task 2: Decimals as Series
A repeating decimal is actually an infinite geometric series. Represent the following repeating decimal as a fraction in simplest form using the sum formula.
0.121212...
1. Break into pieces:
\( 0.12 + 0.0012 + \dots \)
2. Identify variables:
\( a = \) __________
\( r = \) __________
Calculation Zone
Task 3: Zeno's Challenge
A ball is dropped from a height of 10 meters. Every time it hits the ground, it bounces back to 80% of its previous height.
A. Sketch the heights of the first 4 bounces below.
B. Calculate the total vertical distance the ball travels before coming to rest.
(Hint: The ball travels each height twice, except for the first drop!)
Fractal Math Slides FRACTAL MATH
Self-Similarity and Infinite Boundaries
The Area Paradox
Start with an equilateral triangle of area A. Remove the middle triangle. Repeat for the remaining 3 triangles.
Step 0: Area = 1
Step 1: Area = 3/4
Step 2: Area = (3/4)^2
"What is the area as Step \(\to \infty\)?"
The Perimeter Paradox
Growth Pattern
Each segment of length \(L\) is replaced by 4 segments of length \(L/3\).
Ratio \(r = \frac{4}{3}\)
Since \(r > 1\), the perimeter diverges to infinity.
The Contradiction
The Koch Snowflake has:
Infinite Perimeter
Finite Area
Summing the Snowflake Area
Initial Area: \( A_0 \)
Additional area at each step forms a geometric series.
\[ A_{\infty} = A_0 + \sum_{n=1}^{\infty} 3 \cdot 4^{n-1} \left( \frac{1}{9^n} A_0 \right) \]
\( \frac{8}{5} A_0 \)
A finite limit for an infinitely complex shape.
Snowflake Area Project SNOWFLAKE AREA
Project: Modeling the Koch Snowflake
Goal
8/5 A₀
The Koch Snowflake is constructed by starting with an equilateral triangle. In each iteration, the middle third of every segment is replaced by two sides of an equilateral triangle pointing outwards.
Objective: Prove that while the perimeter is infinite, the total area is exactly 1.6 times the original triangle's area.
1
Analyze the First Iteration
In the first step, one new triangle is added to each of the 3 original sides. The side length of these new triangles is 1/3 of the original side length \(s\).
Recall: The area of an equilateral triangle is \( \frac{\sqrt{3}}{4}s^2 \).
Area of New Triangle
\( A_{new} = \frac{1}{9} A_0 \)
Total Added Area (Step 1)
\( 3 \times \frac{1}{9} A_0 = \frac{1}{3} A_0 \)
2
Define the Geometric Series
In each subsequent step, the number of new triangles added increases by a factor of 4, while the area of each new triangle decreases by a factor of 9.
Derive the Series Parameters
First term of added areas (\(a\))
Common ratio of added areas (\(r\))
3
The Final Calculation
Calculate the sum of the infinite geometric series of added areas, then add it to the initial area \(A_0\).
Show Formal Proof Here
Future Value Slides WEALTH FORMULAS
Mathematics of Compound Growth
The Power of Time
Investor A
Invests $2,000 every year starting at age 20. Stops at age 30. (10 years of investing)
Investor B
Invests $2,000 every year starting at age 30. Continues until age 65. (35 years of investing)
Who has more at retirement?
(Assume 7% annual growth for both)
What is an Annuity?
An annuity is a sequence of equal payments made at regular intervals. Think: monthly savings, 401(k) contributions, or insurance payouts.
Future Value
How much will your account be worth after \(n\) payments?
The Model
Sum of future values of each deposit:
\( P(1+i)^{n-1} \) +
This is a geometric series!
The Annuity Formula
\[ FV = P \left[ \frac{(1+i)^n - 1}{i} \right] \]
P
Payment Amount
i
Interest Rate
n
Number of Periods
Fortune Finder Worksheet FORTUNE FINDER
Unit: Financial Series Modeling
Portfolio Holder
Date Opened
Annuity Future Value
For regular payments \(P\) at interest rate \(i\) for \(n\) periods.
\[ FV = P \cdot \frac{(1+i)^n - 1}{i} \]
01 The Rainy Day Fund
Sarah decides to save $150 per month in an account that earns 6% annual interest compounded monthly.
Variables
\( P = \) $150
\( i = 0.06 / 12 = \) _______
\( n = 5 \text{ years} \times 12 = \) _______
Calculate the value after 5 years:
Final Total: $_________________
02 The Retirement Race
Compare two retirement scenarios. Both accounts earn 8% annually.
Plan A: Early Starter
Invest $3,000/year for 10 years (ages 25-35), then stop and let it sit until age 65.
Step 1: FV at age 35
Step 2: Compound Value at 65
Plan B: Late Bloomer
Invest $3,000/year for 30 years (ages 35-65).
Step 1: FV at age 65
"Which plan yields more? Why does the geometric series favor the early starter?"
Loan Logic Slides Credit Calculus
Loans, Amortization, and Present Value
The Price of Today
A loan is the Present Value of a series of future payments.
The bank gives you money now, and you repay it over time. The sum of the "worth" of all those future payments must equal the amount you borrowed today.
Present Value (PV)
What a future sum is worth in today's dollars.
Future vs Present
Value at end: FV
Value now: PV = \frac{FV}{(1+i)^n}
Amortization Formula
To find the monthly payment (\(P\)) for a loan amount (\(PV\)):
\[ P = PV \left[ \frac{i(1+i)^n}{(1+i)^n - 1} \right] \]
Wait, is this a geometric series?
Yes! It is derived by setting the loan amount equal to the sum of discounted payments: \( PV = \sum_{k=1}^n \frac{P}{(1+i)^k} \)
Key Factor
As interest (\(i\)) or time (\(n\)) increases, the portion of your payment going toward interest increases dramatically.
CAR DEAL ALERT
MSRP $35,000
Offer 0% APR for 60 mos
vs. $3,000 cash back at 4% APR
The Math of Choice
Dealers use these numbers to manipulate "monthly affordability."
Project Task: You will use the Present Value formula to prove which deal is actually cheaper.
Calculation Time!
Car Loan Project Worksheet THE LOAN LAB
Summative Project: Amortization Analysis
BORROWER:
CREDIT SCORE:
EXCELLENT
Case Study: The SUV Dilemma
You are buying a car for $42,000. The dealership offers two competing financing options. Your task is to use the present value of a geometric series to analyze which is the better financial decision.
Deal A: The 0% Trap
Sale Price: $42,000
APR: 0.0%
Term: 60 Months
Monthly Payment
Deal B: The Cash Rebate
Sale Price: $37,500
($4,500 Instant Rebate)
APR: 4.5%
Term: 60 Months
Monthly Payment
1. Derive the Periodic Interest Rate
Convert the annual interest rate of Deal B to a monthly decimal (\(i\)).
2. Calculate the Monthly Payments for Deal B
Use the formula: \( P = PV \left[ \frac{i(1+i)^n}{(1+i)^n - 1} \right] \)
Step-by-step substitution required
3. The Cost of Debt
Calculate the Total Interest Paid for Deal B over the life of the loan. Is Deal B still cheaper than Deal A's sticker price?
Executive Summary
Based on your calculations, which deal would you recommend to a buyer? Justify your answer using the concept of present value and the total cost of financing.
Total Cost of Deal A
$_______________
Total Cost of Deal B
$_______________
