Introduction to Modelling with Sequences and Series
Welcome! In this chapter, we are going to explore how the sequences and series you have been studying can be used to describe the real world. This is called modelling. Whether you are calculating how much money you will have in a savings account after ten years or predicting how a population of red squirrels might grow, you are using mathematical models.
Don't worry if "modelling" sounds a bit intimidating. It basically means taking a real-life situation and choosing the right mathematical "tool" (like an arithmetic or geometric sequence) to solve it. Let's break it down!
1. Arithmetic Models: Constant Change
An Arithmetic Sequence is the perfect model when something increases or decreases by the same amount every time. This is often called "linear growth" or "linear decay."
Real-World Example: Imagine you start with £50 in a piggy bank at Week 1 and add £10 every single week.
• Your starting value (\( a \)) is 50.
• Your weekly increase (\( d \), the common difference) is 10.
• Week 1 (Start, \( u_1 \)): £50
• Week 2 (\( u_2 \)): £60
• Week 3 (\( u_3 \)): £70
Key Formula Recap:
To find the value at term \( n \), use: \( u_n = a + (n-1)d \)
To find the total amount saved over \( n \) weeks, use the sum formula: \( S_n = \frac{n}{2}(2a + (n-1)d) \) or \( S_n = \frac{n}{2}(a + l) \) where \( l \) is the last term.
Quick Tip: Remember, if the value is going down (like a leaking water tank), your common difference \( d \) will be a negative number!
Key Takeaway: Use arithmetic models when the change is a fixed amount (e.g., adding £5, losing 2 litres).
2. Geometric Models: Percentage Change
A Geometric Sequence is used when something changes by a percentage or a ratio. This is very common in nature and finance.
Real-World Example (Growth): A house increases in value by 5% every year.
• To find the next year's value, you multiply by 1.05.
• Here, your common ratio (\( r \)) is 1.05.
Real-World Example (Decay): A car loses 20% of its value every year.
• If it loses 20%, it keeps 80%.
• To find the next year's value, you multiply by 0.8.
• Your common ratio (\( r \)) is 0.8.
Key Formula Recap:
To find the value at time \( n \): \( u_n = ar^{n-1} \)
To find the total value (e.g., total distance a bouncy ball travels): \( S_n = \frac{a(1-r^n)}{1-r} \)
For an infinite convergent series where \( |r| < 1 \), the sum to infinity is: \( S_\infty = \frac{a}{1-r} \)
Did you know? Compound interest is just a geometric sequence! This is why saving even small amounts early in life can lead to huge sums later on due to the "multiplicative" effect.
Key Takeaway: Use geometric models when the change is a percentage or proportion (e.g., doubling, tripling, or increasing by 10%).
3. Using Recurrence Relations in Modelling
Sometimes, a situation is better described by how one stage leads to the next. This is a recurrence relation, written as \( a_{n+1} = f(a_n) \).
Real-World Example: A fish farmer has 1000 fish. Every year, the population increases by 10%, but the farmer harvests 200 fish.
• The "increase by 10%" part is \( 1.1 \times a_n \).
• The "harvest 200" part is \( -200 \).
• The model is: \( a_{n+1} = 1.1a_n - 200 \), with \( a_1 = 1000 \).
Common Mistake to Avoid: When using your calculator for these, make sure you keep track of which term is which. \( a_1 \) is your starting value. Students often accidentally use the starting value as \( a_0 \), which can shift your final answer by one year/term!
Key Takeaway: Recurrence relations are great for "step-by-step" models where multiple things happen in each time period.
4. Critical Thinking: Limitations and Refinements
In the OCR MEI syllabus, you aren't just expected to do the math; you need to comment on whether the model is realistic. No mathematical model is perfect!
Common Limitations to look for:
• Population Growth: A geometric model suggests a population will grow forever. In reality, animals run out of food or space (limitations).
• Financial Models: Interest rates rarely stay the same for 20 years (assumptions).
• Discrete vs Continuous: Sequences move in steps (1, 2, 3...). If something changes continuously (like a plant growing every second), a sequence might only be an approximation.
How to "Refine" a model:
If a model isn't working well, we refine it. For example, if our fish pond model predicts "negative fish" (which is impossible!), we might refine the model to say \( a_{n+1} = 0 \) if the calculation drops below zero.
Key Takeaway: Always ask yourself: "Does this answer make sense in the real world?" If your model says a car is worth £0.01 after 5 years, the model is probably wrong!
Summary Review Box
1. Arithmetic: Fixed amount added/subtracted. Use \( u_n = a + (n-1)d \) and \( S_n = \frac{n}{2}(2a + (n-1)d) \).
2. Geometric: Percentage or ratio change. Use \( u_n = ar^{n-1} \) and \( S_n = \frac{a(1-r^n)}{1-r} \).
3. Summing up: If the question asks for the total over time (total distance, total savings), use the \( S_n \) formulas or sigma (\( \sum \)) notation.
4. Convergence: If a geometric model has \( |r| < 1 \), the sum to infinity is \( S_\infty = \frac{a}{1-r} \) (like calculating the total theoretical distance a bouncing ball travels before stopping).
5. Reality Check: Always consider if the model's assumptions (like constant interest rates) are realistic.
Keep practicing! Modelling is often the "wordiest" part of the exam, but once you identify if it is Arithmetic or Geometric, the math is exactly what you have already mastered.