MAT135 Week 2: Unveiling Patterns - Part 1

Welcome back to Professor Baker's Math Class! This week, we embarked on an exciting journey into the realm of patterns, covering sections 2.1 and 2.2 of our textbook. We explored how patterns are fundamental to mathematics and how recognizing them can simplify complex problems. Let's recap the key concepts we covered:

Identifying and Describing Patterns

At the heart of understanding patterns lies the ability to identify them. We discussed various types of patterns, including:

  • Numerical Patterns: Sequences of numbers that follow a specific rule. For example, the sequence 2, 4, 6, 8,... increases by 2 each time. We can represent this pattern algebraically.
  • Geometric Patterns: Arrangements of shapes that repeat or transform according to a rule. Think of tessellations or fractal patterns!
  • Repeating Patterns: A core set of elements that repeat in a predictable order (e.g., A-B-C, A-B-C, A-B-C...).
  • Growing Patterns: Patterns that change over time according to a specific rule (e.g., a plant growing taller each day).

A key skill is expressing these patterns mathematically. For the numerical pattern 2, 4, 6, 8,... we can express the $n^{th}$ term as $a_n = 2n$.

Strategies for Finding Patterns

We also looked at strategies for finding patterns when they aren't immediately obvious:

  1. Look for differences: Calculate the differences between consecutive terms in a sequence. If the differences are constant, you have an arithmetic sequence. If the differences of the differences are constant, it's a quadratic sequence, and so on.
  2. Look for ratios: Calculate the ratios between consecutive terms. If the ratios are constant, you have a geometric sequence.
  3. Visualize: Draw diagrams or charts to help you see the pattern more clearly. This is especially helpful for geometric patterns.
  4. Try to generalize: Once you think you've found a pattern, try to express it as a general rule or formula.
  5. Test your rule: Make sure your rule works for all the terms in the sequence or pattern.

Arithmetic and Geometric Sequences

We introduced two important types of sequences:

  • Arithmetic Sequences: Sequences where the difference between consecutive terms is constant. This constant difference is called the common difference, often denoted as 'd'. The general form of an arithmetic sequence is: $$a_n = a_1 + (n-1)d$$ where $a_n$ is the $n^{th}$ term and $a_1$ is the first term.
  • Geometric Sequences: Sequences where the ratio between consecutive terms is constant. This constant ratio is called the common ratio, often denoted as 'r'. The general form of a geometric sequence is: $$a_n = a_1 * r^{(n-1)}$$

Example Problem

Let's look at an example: Find the next three numbers in the sequence 1, 4, 9, 16,...

By recognizing that these are the squares of consecutive integers (12, 22, 32, 42), we can easily determine the next three terms: 25, 36, and 49.

Remember, practice is key to mastering pattern recognition! Keep an eye out for patterns in your everyday life. You might be surprised at how often they appear. Keep exploring, keep questioning, and keep learning!

As mentioned in class, the project link is available here and the class notes here.