Pattern Pulse Slides Module 01
PATTERN PULSE
Distinguishing Arithmetic and Geometric Behaviors
FINANCIAL & BIOLOGICAL MODELING
The Career Choice
Option A: The Steady Climb
Starting Salary: $60,000
Annual Raise: +$5,000 (Fixed)
Option B: The Percent Push
Starting Salary: $45,000
Annual Raise: +8% (Variable)
Which offer is better after 5 years? What about 20 years?
The DNA of Change
Arithmetic
A sequence where the difference between consecutive terms is constant.
\( a_n = a_{n-1} + d \)
Recursive Definition
Geometric
A sequence where the ratio between consecutive terms is constant.
\( g_n = g_{n-1} \cdot r \)
Recursive Definition
Breaking the Chain: Explicit Formulas
Arithmetic
\( a_n = a_1 + (n-1)d \)
Allows you to find any term without knowing the one before it. Essential for long-term forecasting.
Geometric
\( g_n = g_1 \cdot r^{n-1} \)
The power of exponentiation. Note how the growth accelerates as \( n \) increases.
Identification Lab
Case 001
8, 12, 18, 27...
Find the behavior and the next term.
Case 002
102, 95, 88, 81...
Find the behavior and the 50th term.
Case 003
5, 5, 5, 5...
Wait—is this both or neither?
Case 004
\( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \)...
Define explicitly.
Sequence Shifter Worksheet Sequence Shifter
Pattern Identification & Formula Derivation
Name:
Date:
1
Classification Lab
Determine if each sequence is arithmetic, geometric, or neither. If it is arithmetic, state the common difference (d). If it is geometric, state the common ratio (r).
2, 6, 18, 54, ...
Type:
d or r:
100, 93, 86, 79, ...
Type:
d or r:
1, 4, 9, 16, ...
Type:
d or r:
2
The Conversion Suite
Convert the following recursive definitions into explicit formulas.
\( a_1 = 12 \)
Write explicit formula here
\( g_1 = 5 \)
Write explicit formula here
3
The Analyst's Edge
Scenario: A startup offers you a contract starting at $50,000. You have two options for your annual salary increase:
Plan A: Fixed increase of $3,500 per year.
Plan B: Fixed increase of 6% per year.
Write the explicit formula for Plan A and calculate the salary for Year 10.
Write the explicit formula for Plan B and calculate the salary for Year 10.
Sequence Shifter Key Answer Key
Sequence Shifter | Pattern Identification
1. Classification Lab
2, 6, 18, 54, ...
Geometric
r = 3
100, 93, 86, 79, ...
Arithmetic
d = -7
1, 4, 9, 16, ...
Neither
N/A (Perfect Squares)
2. The Conversion Suite
\( a_1 = 12, a_n = a_{n-1} - 4.5 \)
\( a_n = 12 + (n-1)(-4.5) \)
OR \( a_n = 16.5 - 4.5n \)
\( g_1 = 5, g_n = 3(g_{n-1}) \)
\( g_n = 5 \cdot 3^{n-1} \)
3. The Analyst's Edge
Plan A (Arithmetic):
\( a_n = 50,000 + (n-1)3,500 \)
For Year 10 (\( n=10 \)): \( 50,000 + (9 \cdot 3,500) = 50,000 + 31,500 = \mathbf{\$81,500} \)
Plan B (Geometric):
\( g_n = 50,000 \cdot (1.06)^{n-1} \)
For Year 10 (\( n=10 \)): \( 50,000 \cdot (1.06)^9 \approx 50,000 \cdot 1.68947 = \mathbf{\$84,473.50} \)
Linear Lifespan Slides Module 02
Linear Lifespans
Arithmetic Modeling in Industry & Finance
Simple Interest & Depreciation
The Machinery Problem
In industrial accounting, assets often lose value at a constant rate per month or year.
Straight-Line Depreciation
• Purchase Price: $1.2M
• Monthly Loss: $15,000
• Residual Value: $200,000
Simple Interest = Arithmetic
The Concept
Interest is only earned on the original principal. It never grows on itself.
\( I = P \cdot r \cdot t \)
The Sequence
The value of the investment after \( n \) periods forms an arithmetic sequence.
\( a_n = P + n(Pr) \)
where \( d = Pr \)
Solving for Time
If we know the starting value (\( a_1 \)), the rate of change (\( d \)), and the target value (\( a_n \)), we can solve for \( n \):
\( n = \frac{a_n - a_1}{d} + 1 \)
Essential for finding the "Lifespan" of an asset.
Predictive Analysis Challenge
A fleet of delivery drones costs $500,000 . The resale value drops by $8,000 every month of operation.
1
Write the explicit formula for the value after \( n \) months.
2
In which month will the fleet be worth exactly $116,000?
Machinery Meltdown Case Study Machinery Meltdown
Asset Depreciation Case Study
Unit: Arithmetic Ops
Analyst:
The Briefing: Heavy Industry Corp
Heavy Industry Corp (HIC) has just installed a new high-precision CNC Milling Machine. The machine's performance degrades linearly with use, leading to a fixed reduction in its market value every quarter. You have been tasked with auditing the asset's lifespan and determining when it reaches its "scrap value."
Initial Cost $2,400,000
Quarterly Loss $85,000
Scrap Threshold $360,000
PHASE 01
The Explicit Model
Write an explicit formula for the value of the machine \( V_n \) where \( n \) is the number of quarters passed since installation. (Note: n = 1 represents the end of the first quarter).
Using your formula, calculate the machine's value at the end of Year 3 (8 quarters).
PHASE 02
Terminal Lifespan Analysis
HIC policy dictates that an asset should be sold for scrap the exact quarter its value drops below or hits the Scrap Threshold ($360,000). Show your algebraic work to determine which quarter this machine will be scrapped.
PHASE 03
Financial Projection
"If we could reduce quarterly wear-and-tear by just 10% through better maintenance, how many additional quarters of use would we gain before hitting the scrap value?"
Machinery Meltdown Key Audit Key
Machinery Meltdown | Answer Solutions
PHASE 01
The Explicit Model
Formula:
\( V_n = 2,400,000 - 85,000n \)
*Note: If using \( a_n = a_1 + (n-1)d \), where \( a_1 = 2,315,000 \), result is the same.
Value at end of Year 3 (n = 12 quarters):
\( V_{12} = 2,400,000 - 85,000(12) \)
\( V_{12} = 2,400,000 - 1,020,000 = \mathbf{\$1,380,000} \)
PHASE 02
Terminal Lifespan Analysis
Solving for n when \( V_n \leq 360,000 \):
\( 360,000 = 2,400,000 - 85,000n \)
\( -2,040,000 = -85,000n \)
\( n = \frac{-2,040,000}{-85,000} \)
\( \mathbf{n = 24} \)
Conclusion: The machine will hit the scrap threshold exactly at the end of the 24th quarter (End of Year 6).
PHASE 03
Financial Projection
New Quarterly Loss (d):
\( 85,000 \cdot 0.90 = \$76,500 \)
New Lifespan calculation:
\( 360,000 = 2,400,000 - 76,500n \)
\( n = \frac{2,040,000}{76,500} \approx 26.66 \)
Gains: Approximately 2.66 quarters. In practical terms, HIC would gain 2 full quarters of production (extending to 26 quarters total).
Exponential Edge Slides Module 03
Exponential Edge
Geometric Progressions & Compound Growth
Viral Spread & Finance
The Paper Fold Paradox
If you could fold a standard piece of paper in half 42 times, how thick would it be?
A) As thick as a phone book?
B) As tall as a skyscraper?
C) It would reach the Moon?
Geometric sequences expand beyond human intuition.
Answer: C
~ 384,400 km
The Anatomy of a Ratio
Common Ratio (r)
The factor by which we multiply each term to get the next.
Growth r > 1
Decay 0 < r < 1
Compound Growth
Interest earning interest. Viral infections spreading through a population.
\( g_n = g_1 \cdot r^{n-1} \)
Viral Spread Analysis
If one person infects 3 others daily, and each of them infects 3 others, the sequence of "newly infected" is:
1, 3, 9, 27, 81...
Calculating the Nth Day
\( g_n = 1 \cdot 3^{n-1} \)
On Day 10
19,683 new cases
The Next Step: Summation
If we want to know the total number of people infected from Day 1 to Day 10, we don't just need the sequence...
We need the GEOMETRIC SERIES.
Coming in Lesson 04
Viral Velocity Activity Viral Velocity
Geometric Growth Simulation
Topic: Exponential Modeling
Student:
Phase 1: Propagation Mechanics
In a biological model, the Basic Reproduction Number (\(R_0\)) represents the number of cases one case generates on average over the course of its infectious period. This is the common ratio (r) of our geometric sequence.
Scenario Alpha
Patient Zero arrives in a city. For this pathogen, the \(R_0\) is 4. Every transmission cycle (1 week), each infected person transmits to 4 new people.
1. List the first 5 terms of the sequence (New Cases per week):
1,
2. Define the explicit formula for Scenario Alpha:
Scenario Beta
With public health measures, the \(R_0\) is reduced to 1.5. Patient Zero still starts the sequence.
3. List the first 5 terms of the sequence (Round to 2 decimals):
1,
4. Define the explicit formula for Scenario Beta:
Phase 2: Impact Analysis
Calculate the number of NEW cases in Week 12 for both scenarios. Show your work.
Scenario Alpha (R0 = 4)
Scenario Beta (R0 = 1.5)
Critical Thinking Question:
The healthcare system collapses if new cases in a single week exceed 100,000. In which week does Scenario Alpha collapse the system? What about Scenario Beta?
Viral Velocity Key Analyst Guide
Viral Velocity | Solutions Key
Phase 1: Propagation Mechanics
Scenario Alpha (R0 = 4)
1, 4, 16, 64, 256
Explicit Formula:
\( g_n = 1 \cdot 4^{n-1} \) OR \( g_n = 4^{n-1} \)
Scenario Beta (R0 = 1.5)
1, 1.5, 2.25, 3.38, 5.06
Explicit Formula:
\( g_n = 1 \cdot (1.5)^{n-1} \)
Phase 2: Impact Analysis
Alpha Week 12:
\( g_{12} = 4^{11} = \mathbf{4,194,304} \)
This is a massive catastrophe.
Beta Week 12:
\( g_{12} = (1.5)^{11} \approx \mathbf{86.50} \)
Highly manageable.
Threshold Calculation (System Collapse):
Scenario Alpha:
\( 100,000 < 4^{n-1} \)
\( \log(100,000) < (n-1)\log(4) \)
\( 5 / 0.602 < n-1 \)
\( 8.3 < n-1 \Rightarrow n > 9.3 \)
Collapse in Week 10.
Scenario Beta:
\( 100,000 < 1.5^{n-1} \)
\( \log(100,000) / \log(1.5) < n-1 \)
\( 11.51 / 0.405 \approx 28.4 \)
\( 28.4 < n-1 \Rightarrow n > 29.4 \)
Collapse in Week 30.
Note: Reducing R0 buys critical time for healthcare response.
Future Fortune Slides Module 04
Future Fortunes
The Power of Geometric Series in Finance
Annuities & Loan Mastery
From Sequence to Series
Sequence
A list of separate payments over time.
\( a, ar, ar^2, ar^3 \)...
Series
The total sum of all those payments.
\( S_n = a + ar + ar^2 + ... + ar^{n-1} \)
The Finite Sum Formula
\( S_n = a \frac{1 - r^n}{1 - r} \)
This formula is the engine behind every modern financial instrument.
Financial Engineering
The Annuity
A sequence of equal payments made at regular intervals.
• Monthly 401(k) contributions
• Mortgage payments
• Student loan repayments
Two Values
Future Value (FV)
How much your savings will be worth later.
Present Value (PV)
What a series of future payments is worth today.
The Wealth Builder
"If I save $500 every month at 6% annual interest, how much do I have after 30 years?"
Rate (Monthly) 1.005
Periods (n) 360
Payment (a) $500
\( FV = 500 \frac{1.005^{360} - 1}{0.005} \)
≈ $502,257.52
Loan Officer Project
You are now the lead analyst for a high-end real estate lender. Your job is to calculate mortgage structures for multi-million dollar properties using the present value of geometric series.
Begin Simulation →
Mortgage Mastery Project Mortgage Mastery
Complex Financial Modeling Project
Project Code: FIN-302
Loan Officer Certification
Analyst:
The High-Stakes Close
Project Brief: The Skyline Estate
A client is purchasing a $1,500,000 penthouse. They have a $300,000 down payment and need to finance the remaining balance. You must calculate the monthly mortgage payment using the Present Value of a Geometric Series.
Loan Terms
• Loan Amount: $1,200,000
• Annual Interest Rate: 4.8%
• Loan Duration: 30 Years
Formula Reference
Present Value of Annuity:
\( PV = Pmt \cdot \frac{1 - (1+i)^{-n}}{i} \)
Where \( i \) is monthly rate and \( n \) is total months.
01
Variable Derivation
Monthly Interest (\( i \))
Total Periods (\( n \))
Loan Principal (\( PV \))
02
The Payment Calculation
Rearrange the formula to solve for \( Pmt \). Show every step of your algebraic work.
03
Total Interest Analysis
Calculate the total amount paid over 30 years:
Calculate the total interest paid to the bank:
"Does it surprise you that the total interest is nearly as much as (or more than) the original loan? Explain why this happens in a geometric series model."
Mortgage Mastery Key Director's Key
Mortgage Mastery | Project Solutions
01. Variable Derivation
Monthly Rate (i)
\( 0.048 / 12 = \mathbf{0.004} \)
Total Periods (n)
\( 30 \cdot 12 = \mathbf{360} \)
Loan Principal (PV)
\(\mathbf{\$1,200,000} \)
02. The Payment Calculation
Rearranged Formula: \( Pmt = \frac{PV \cdot i}{1 - (1+i)^{-n}} \)
\( Pmt = \frac{1,200,000 \cdot 0.004}{1 - (1.004)^{-360}} \)
\( Pmt = \frac{4,800}{1 - 0.23788} \)
\( Pmt = \frac{4,800}{0.76212} \)
\( Pmt = \mathbf{\$6,298.22} \)
03. Total Cost Analysis
Total Paid over 30 Years:
\( 6,298.22 \cdot 360 = \mathbf{\$2,267,359.20} \)
Total Interest Paid:
\( 2,267,359.20 - 1,200,000 = \mathbf{\$1,067,359.20} \)
Qualitative Answer:
"The total interest is high because compounding works in reverse on a debt. In the early years of a loan, the principal is high, so the 0.4% monthly interest applies to a large number, consuming most of the monthly payment and leaving little to reduce the principal itself. This 'slow start' is a fundamental trait of geometric decay in financial series."
Growth Clash Slides Module 05
Growth Boundaries
Comparing Rates of Change & System Limitations
The Malthusian Debate
Arithmetic vs. Geometric
The Hare (Geometric)
Starts slow, but the rate of change is proportional to its size. It will eventually overtake any arithmetic growth, no matter how large the starting difference.
\( y = a \cdot r^x \)
The Tortoise (Arithmetic)
The rate of change is constant. Reliable and predictable, but fundamentally slower in the long run.
\( y = m \cdot x + b \)
The Malthusian Catastrophe
"The power of population is indefinitely greater than the power in the earth to produce subsistence for man."
— Thomas Malthus, 1798
Population Geometric Growth
Food Supply Arithmetic Growth
Why don't we see infinite growth?
Resource Scarcity
Models assume infinite space, water, and fuel.
Carrying Capacity
Environmentally defined upper limits (Plateauing).
Saturation
Market or biological saturation points.
Mathematical models are tools, not destinies.
The Policy Debate
If a population is growing at 3% per year (Geometric) and its food supply is increasing by 1,000 tons per year (Arithmetic), what policy interventions should be prioritized to prevent a crisis?
Critical Discussion
Malthusian Limit Worksheet The Malthusian Limit
Comparative Analysis & Modeling
Case ID: 2026-BIO
Researcher:
The Simulation Brief
A remote island colony is being monitored. You must model two competing variables over 10 cycles (decades) to predict the point of "Resource Exhaustion."
Population (Geometric)
Starts at 1,000 people and grows by 50% every decade.
\( P_n = 1000(1.5)^n \)
Food Units (Arithmetic)
Starts at 5,000 units and increases by 4,000 units every decade.
\( F_n = 5000 + 4000n \)
Cycle Data Table
Cycle (n) Population (\( P_n \)) Food Units (\( F_n \)) Status (Surplus/Deficit) 0 1,000 5,000 Surplus 1 2 3 4 5 6
Critical Evaluation
1. Intersection Point Identification
In which decade does the population first exceed the food supply? Show the comparison from your table.
2. Model Criticism
In reality, will the population actually reach the levels calculated in cycle 10? Why or why not? Mention "carrying capacity" in your response.
3. Innovation Variable
If a technological breakthrough changes the food supply from an Arithmetic sequence to a Geometric sequence with \( r = 1.4 \), how does the long-term outlook change?
Malthusian Limit Key Verification Key
The Malthusian Limit | Solved Simulation
Cycle Data Results
Cycle (n) Population (\( P_n \)) Food Units (\( F_n \)) Status 0 1,000 5,000 Surplus 1 1,500 9,000 Surplus 2 2,250 13,000 Surplus 3 3,375 17,000 Surplus 4 5,062.5 21,000 Surplus 5 7,593.75 25,000 Surplus 6 11,390.6 29,000 Surplus 7 17,085.9 33,000 Surplus 8 25,628.9 37,000 Surplus 9 38,443.3 41,000 Surplus 10 57,665.0 45,000 DEFICIT
Critical Evaluation Key
1. Intersection Point Identification:
The population first exceeds the food supply in Cycle 10. Note: While Cycle 9 is close, the surplus remains until Cycle 10 when the population hits ~57.6k against 45k food units.
2. Model Criticism:
"The population is unlikely to reach cycle 10 levels because it will hit the island's carrying capacity. As resources become scarce, the growth rate will naturally decrease (shifting from a geometric sequence to a logistic model), or mortality rates will increase, plateauing the population before the crash occurs."
3. Innovation Variable:
If food also grows geometrically (\( r = 1.4 \)), it still loses to the population (\( r = 1.5 \)), but the crash happens much later. However, if the food growth ratio were higher than the population growth ratio, the scarcity would be permanently avoided.
