Sequences (applications) - Mathematics: Applications & Interpretation IB Study Notes
Overview
Imagine you're saving money, or watching a plant grow, or even seeing how a rumour spreads through your school. All these things follow a pattern over time. Sometimes the pattern adds the same amount each time, and sometimes it multiplies by the same amount. This topic is all about understanding these patterns, called 'sequences', and using them to predict what will happen in the future. Why is this important? Because the real world is full of these patterns! From calculating how much interest your savings account earns to figuring out how populations of animals change, sequences help us make sense of the world and plan for what's next. It's like having a crystal ball, but for numbers! We'll learn about two main types of sequences: **arithmetic sequences**, where you add or subtract the same number each time (like counting up by 2s), and **geometric sequences**, where you multiply or divide by the same number each time (like doubling your money). We'll also see how to add up all the numbers in a sequence, which is super useful for things like total savings over many years.
What Is This? (The Simple Version)
Think of a sequence like a list of numbers that follow a specific rule. It's like a recipe where each step tells you how to get the next ingredient. For example, if you start with 2 and your rule is "add 3", your sequence would be 2, 5, 8, 11, and so on.
In this topic, we're looking at how these number patterns show up in real life and how we can use math to understand them. We'll focus on two main types:
- Arithmetic Sequences: Imagine you get 5 extra minutes of screen time every day. Day 1: 30 minutes, Day 2: 35 minutes, Day 3: 40 minutes. You're adding the same amount (+5 minutes) each time. This constant amount is called the common difference.
- Geometric Sequences: Imagine a magic bean that doubles its height every hour. Hour 1: 2cm, Hour 2: 4cm, Hour 3: 8cm. You're multiplying by the same amount (x2) each time. This constant multiplier is called the common ratio.
We're not just listing numbers; we're learning to predict the 100th number, or how much you'd have saved after 20 months, without having to list every single one!
Real-World Example
Let's say your aunt gives you $100 for your birthday, and she promises to add $20 to that amount every year for your next birthdays. You want to know how much money she'll have given you by your 18th birthday.
Let's break it down:
- Your 13th Birthday (first birthday after the initial gift): You get the initial $100 + $20 = $120.
- Your 14th Birthday: You get another $20, so $120 + $20 = $140.
- Your 15th Birthday: You get another $20, so $140 + $20 = $160.
See the pattern? Each year, the amount you receive for that birthday is increasing by $20. This is an arithmetic sequence because we are adding the same amount each time. The first term (the amount for your 13th birthday) is $120, and the common difference (the amount added each year) is $20.
Now, if we wanted to know the total amount she's given you by your 18th birthday, we'd have to add up all those birthday amounts. That's where the idea of a series (adding up the terms of a sequence) comes in handy!
How It Works (Step by Step)
Let's use the birthday money example to see how we find a specific term or the total sum. **Step 1: Identify the type of sequence.** * Is a fixed amount being added or subtracted? (Arithmetic) * Is a fixed amount being multiplied or divided? (Geometric) * In our birthday example, $20 is a...
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Key Concepts
- Sequence: An ordered list of numbers that follow a specific rule or pattern.
- Term: Each individual number in a sequence.
- Arithmetic Sequence: A sequence where each term after the first is found by adding a constant, called the common difference, to the previous term.
- Common Difference (d): The constant value added or subtracted between consecutive terms in an arithmetic sequence.
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Exam Tips
- →Always identify if a problem is arithmetic or geometric *before* choosing a formula. This is the most crucial first step.
- →Write down the given values (u₁, d or r, n) clearly before attempting to solve. This helps organize your thoughts and reduces errors.
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