Welcome to the World of Patterns and Logic!

Have you ever noticed how the petals of a flower grow, or how a computer knows exactly how to sort your playlist? These aren't just random occurrences; they are built on patterns, sequences, and algorithms. In this chapter, we will learn how to spot these patterns, describe them using mathematical "rules," and follow step-by-step instructions (algorithms) to solve complex problems. Whether you are aiming for the Standard or Extended level, mastering these skills is like learning the secret code of the universe!

1. Understanding Sequences

A sequence is simply a list of numbers that follow a specific order or rule. Each number in the list is called a term. We usually use the symbol \( u_n \) to represent the "\( n \)-th" term (for example, \( u_1 \) is the first term, \( u_2 \) is the second, and so on).

Arithmetic Sequences: Adding and Subtracting

In an arithmetic sequence, the difference between one term and the next is always the same. We call this the common difference (\( d \)).

Example: \( 5, 8, 11, 14, ... \)
Here, we are adding \( 3 \) every time. So, the common difference \( d = 3 \).

The General Rule: To find any term (\( u_n \)), we use the formula:
\( u_n = u_1 + (n - 1)d \)

Memory Aid: Think of "Arithmetic" as "Addition." You are just stepping forward by the same amount each time!

Geometric Sequences: Multiplying and Dividing

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio (\( r \)).

Example: \( 2, 6, 18, 54, ... \)
Here, we are multiplying by \( 3 \) every time. So, the common ratio \( r = 3 \).

The General Rule: To find any term (\( u_n \)), we use the formula:
\( u_n = u_1 \times r^{(n-1)} \)

Quick Tip: If the numbers are getting smaller, your common ratio is a fraction! For example, if you divide by \( 2 \) every time, your ratio is \( r = \frac{1}{2} \).

Key Takeaway: Arithmetic sequences use a constant difference; Geometric sequences use a constant ratio.

2. Investigating Patterns (Criterion B)

In your MYP eAssessments, you will often be asked to "investigate" a pattern. This is where you become a math detective! Follow these steps:

  1. Gather Data: Look at the first few shapes or numbers and organize them into a table.
  2. Find the Relationship: Is it adding a fixed amount? Multiplying? Or maybe it's related to square numbers (\( n^2 \))?
  3. State the Rule: Write a formula (like the ones above) that describes the pattern.
  4. Verify: Use your formula to calculate a term you already know to see if it works.
  5. Justify: Explain why the rule works based on how the pattern grows.

Did you know? The Fibonacci sequence (\( 1, 1, 2, 3, 5, 8... \)) is found everywhere in nature, from pinecones to galaxies! In this sequence, you find the next term by adding the two previous terms together.

3. Algorithms: The "Recipe" for Success

An algorithm is just a fancy word for a well-defined, step-by-step procedure for solving a problem. Think of it like a recipe for baking a cake: if you follow the steps in order, you get the same result every time.

How to Analyze an Algorithm:

When you see an algorithm in math, it might look like a flowchart or a list of instructions. To master them, you should:

  • Trace the steps: Follow the instructions with a specific starting number (input) and see what happens (output).
  • Identify the logic: Look for "if-then" statements (decisions) and "loops" (repeating a step until a condition is met).
  • Efficiency: Ask yourself, "Is there a faster way to get to the answer?"

Example of a simple algorithm:
1. Pick a number (\( x \)).
2. Add \( 5 \).
3. Multiply by \( 2 \).
4. Subtract \( 10 \).
5. Output: The result is always \( 2 \times x \)!

Key Takeaway: Algorithms turn complex problems into a series of manageable steps. They are the foundation of how computers work!

4. Series: Adding it all up (Extended Mathematics Only)

If you are taking Extended Mathematics, you need to know about series. A series is simply the sum of the terms in a sequence.

Arithmetic Series

To find the sum of the first \( n \) terms of an arithmetic sequence (\( S_n \)):
\( S_n = \frac{n}{2}(u_1 + u_n) \)

Geometric Series

To find the sum of the first \( n \) terms of a geometric sequence:
\( S_n = \frac{u_1(r^n - 1)}{r - 1} \) (where \( r \neq 1 \))

Infinite Geometric Series

Sometimes, if a geometric sequence gets smaller and smaller (like \( 10, 5, 2.5, 1.25... \)), you can actually add up all the terms forever and get a single number! This only works if the ratio \( r \) is between \( -1 \) and \( 1 \).
The Formula: \( S_{\infty} = \frac{u_1}{1 - r} \)

Common Mistake: Don't confuse the formula for the term (\( u_n \)) with the formula for the sum (\( S_n \)). Always read the question carefully to see if it asks for a specific number in the list or the total of the list!

5. Summary and Quick Review

Don't worry if this seems like a lot of formulas! Just remember the core ideas:

  • Arithmetic: Constant adding/subtracting (\( d \)).
  • Geometric: Constant multiplying/dividing (\( r \)).
  • Rule-making: Use \( n \) to represent the position of the term.
  • Algorithms: Follow the logic step-by-step.
  • Extended: Summing sequences creates a "Series."

Pro-tip for eAssessments: Always show your working (Criterion C). Even if your final answer is slightly off, showing the method you used to find a pattern or follow an algorithm can still earn you many marks!