Welcome to Modelling with Sequences and Series!
In this chapter, we take everything you have learned about Arithmetic and Geometric sequences and apply it to the real world. Think of mathematical modelling as "translating" a real-life situation—like your bank account growing or a population of bees increasing—into a math formula.
Don't worry if this seems tricky at first! Modelling is just about spotting patterns. Once you know if a situation is adding a fixed amount (Arithmetic) or multiplying by a fixed percentage (Geometric), the math becomes much easier.
1. Arithmetic Modelling (The "Adding" Model)
An Arithmetic Sequence is used when something grows or shrinks by the same amount every time.
The Real-World Connection:
• Simple Interest: If you put \( £100 \) in a bank and they give you \( £5 \) every year based only on your original \( £100 \).
• Fixed Increases: A job that gives you a flat \( £500 \) raise every year.
• Linear Depreciation: A car that loses exactly \( £1,000 \) in value every year.
Key Formula Refresher
To find a specific term: \( u_n = a + (n-1)d \)
To find the total sum: \( S_n = \frac{n}{2}(2a + (n-1)d) \)
Memory Aid: Think of Arithmetic as Adding. If you are adding the same "difference" (\( d \)) every time, it is an Arithmetic model.
Example: The Savings Jar
Example: You start with \( £50 \) in a jar and add \( £10 \) every month. How much is in the jar after 2 years?
Step 1: Identify your variables.
Your starting amount (a) is \( 50 \).
Your common difference (d) is \( 10 \).
The number of terms (n) is \( 24 \) (because 2 years = 24 months).
Step 2: Plug into the formula.
\( u_{24} = 50 + (24 - 1) \times 10 \)
\( u_{24} = 50 + 230 = £280 \).
Quick Review: If the change is a fixed amount, use Arithmetic formulas.
2. Geometric Modelling (The "Percentage" Model)
A Geometric Sequence is used when something grows or shrinks by a percentage or a ratio. This is much more common in finance and biology.
The Real-World Connection:
• Compound Interest: This is the "interest on interest" you get in most bank accounts.
• Population Growth: Bacteria doubling every hour or a town growing by \( 2\% \) per year.
• Reducing-Balance Depreciation: A phone losing \( 20\% \) of its value every year.
Key Formula Refresher
To find a specific term: \( u_n = ar^{n-1} \)
To find the total sum: \( S_n = \frac{a(1-r^n)}{1-r} \)
The "Multiplier" (\( r \)):
• If something grows by \( 5\% \), the multiplier \( r = 1.05 \).
• If something shrinks by \( 5\% \), the multiplier \( r = 0.95 \) (because \( 100\% - 5\% = 95\% \)).
Did you know?
Compound interest is often called the "eighth wonder of the world" because of how fast it grows. Even a small starting amount can become huge over time because the multiplier (\( r \)) is applied to an ever-increasing total!
Example: Investing for the Future
Example: You invest \( £1,000 \) at an interest rate of \( 4\% \) per year. How much will you have at the start of the 10th year?
Step 1: Identify your variables.
Starting amount (a) = \( 1000 \).
Rate of growth = \( 4\% \), so the multiplier (r) = \( 1.04 \).
Term (n) = \( 10 \).
Step 2: Use the term formula.
\( u_{10} = 1000 \times (1.04)^{10-1} \)
\( u_{10} = 1000 \times (1.04)^9 \approx £1,423.31 \).
Quick Review: If the change is a percentage or ratio, use Geometric formulas.
3. Solving Inequalities with Logs
In the OCR exam, you are often asked: "How many years will it take for the investment to exceed \( £5,000 \)?" When the unknown value (\( n \)) is in the power, we must use Logarithms.
Step-by-Step Process:
1. Set up the inequality: \( ar^{n-1} > 5000 \).
2. Divide by \( a \): \( r^{n-1} > \frac{5000}{a} \).
3. Take logs of both sides: \( \ln(r^{n-1}) > \ln(\frac{5000}{a}) \).
4. Bring the power down: \( (n-1) \ln(r) > \ln(\frac{5000}{a}) \).
5. Solve for \( n \).
Common Mistake Alert: When you divide by \( \ln(r) \), if \( r < 1 \) (like in decay problems), \( \ln(r) \) will be a negative number. Remember to flip the inequality sign when dividing by a negative!
4. Modelling Assumptions and Limitations
No model is perfect. When you answer a "Modelling" question, you might be asked to comment on its validity.
Common Assumptions:
• Constant Rates: We assume the interest rate or growth rate stays exactly the same every year.
• Infinite Growth: Geometric models suggest populations grow forever, but in real life, they run out of food or space.
Refining the Model:
If the model doesn't match reality, we "refine" it by changing the value of \( r \) or \( d \) to better fit the observed data.
Summary Takeaways
1. Spot the pattern: Constant amount = Arithmetic; Percentage/Ratio = Geometric.
2. Be careful with \( n \): Check if the question asks for the "end of the year" or "start of the year." This usually changes \( n \) by 1.
3. Use your Logs: Practice using logarithms to find \( n \) when it is an exponent.
4. Check your \( r \): For a \( 3\% \) increase, \( r = 1.03 \). For a \( 3\% \) decrease, \( r = 0.97 \).
Keep practicing! Modelling is just a puzzle where you find the right pieces (\( a, r, d, n \)) and put them into the right formula.