Welcome to the World of Patterns!

Have you ever looked at a string of numbers and felt like they were a secret code? In Mathematics, investigating patterns is exactly like being a detective. Your job is to find the hidden "logic" or "rule" that connects the numbers. This chapter focuses on Criterion B: Investigating Patterns, where you will learn how to select the right mathematical tools to describe how a pattern works.

Don't worry if this seems tricky at first! Pattern investigation isn't about being a human calculator; it's about being observant and organized. Once you know which technique to use, the "code" usually cracks itself.

1. Organizing Your Investigation

Before you can find a rule, you need to see your data clearly. The best way to start any investigation is by creating a table of values. This is a key part of Criterion C (Communicating).

Imagine you are building a tower of blocks. Term 1 (\( n = 1 \)) has 3 blocks, Term 2 (\( n = 2 \)) has 5 blocks, and Term 3 (\( n = 3 \)) has 7 blocks. Organizing it looks like this:

  • Position (\( n \)): 1, 2, 3, 4...
  • Value (\( u_n \)): 3, 5, 7, 9...

Quick Tip: Always use \( n \) to represent the position in the sequence (the "step number") and \( u_n \) or \( f(n) \) to represent the actual value at that step.

2. Selecting the Right Technique

How do you know what kind of rule you are looking for? We look at the differences between the numbers. This is the most important "technique" you will select.

Technique A: Finding the First Difference (Linear Patterns)

If the value increases or decreases by the same amount every time, it is a linear pattern.

Example: 4, 7, 10, 13...
The difference between each number is \( +3 \). Because this "first difference" is constant, the rule will look like a straight-line equation: \( f(n) = mn + c \).

Technique B: Finding the Second Difference (Quadratic Patterns)

If the first differences are not the same, don't panic! Check the difference between the differences. This is called the second difference.

Example: 2, 5, 10, 17...
First differences: \( 3, 5, 7 \)
Second differences: \( 2, 2 \)
Because the second difference is constant, this is a quadratic pattern. The rule will involve \( n^2 \).

Technique C: Finding a Constant Ratio (Geometric Patterns - Extended)

For Extended Mathematics students, sometimes numbers don't add or subtract; they multiply. This is a geometric sequence.

Example: 3, 6, 12, 24...
Here, we are multiplying by 2 each time (\( \text{ratio} = 2 \)). The rule will involve an exponent: \( u_n = a \cdot r^{(n-1)} \).

Key Takeaway: Always check the first difference first. If that's not constant, check the second difference. If neither works, check if the numbers are being multiplied!

3. Describing the General Rule

A "general rule" is a formula that allows you to find the value of any term without having to list all the ones before it. It’s like having a teleportation device to the 100th step!

The Linear Rule Formula

If your first difference is \( d \), your rule will start with \( d \cdot n \).
Step 1: Find the difference (\( d \)).
Step 2: Multiply \( d \) by the first position (\( n = 1 \)).
Step 3: See what you need to add or subtract to get the actual first value.

Example: 5, 8, 11, 14...
The difference is \( 3 \). So, start with \( 3n \).
When \( n = 1 \), \( 3(1) = 3 \). But our first value is 5. We need to add 2.
General Rule: \( f(n) = 3n + 2 \)

Mathematical Notation

In the MYP, you should use proper function notation to describe your rules. You can write it as \( f(n) = 3n + 2 \) or \( f: n \rightarrow 3n + 2 \). This shows clear mathematical communication!

Did you know? The "zero term" (\( n = 0 \)) is a secret trick. In a linear pattern, the value of the "step 0" is always the constant \( c \) in your \( f(n) = dn + c \) rule!

4. Common Mistakes to Avoid

  • Confusing \( n \) and \( u_n \): Remember, \( n \) is just the counter (1st, 2nd, 3rd...). Don't accidentally use the sequence values as your \( n \).
  • Stopping too early: Don't just say "the rule is add 2." That is a recursive description. A general rule must work for the n-th term.
  • Not testing the rule: Always test your rule with the 2nd or 3rd term to make sure it actually works! (We will cover more on verifying in the next chapter).

Quick Review

Linear: Constant 1st difference. Rule: \( f(n) = dn + c \)
Quadratic: Constant 2nd difference. Rule: contains \( n^2 \)
Geometric (Extended): Constant multiplier (ratio). Rule: contains \( r^n \)
Goal: Move from a table of values to a general rule using \( n \).

Note: For more information on proving your rules are correct, see the chapter on "Verifying and Justifying General Rules".