You can write a pattern rule using a variable by identifying the relationship between the terms in a sequence and expressing it algebraically. This is done by finding the starting point and the consistent change or pattern, then representing the variable with a letter like n.
Understanding Pattern Rules
Pattern rules describe how terms in a sequence are generated. They can be expressed in words or, more powerfully, using variables. The reference video highlights that we can describe a pattern like 7, 8, 9, 10, 11 as "going up one more in the previous term" or "start at seven add one each time". Using a variable allows us to express this more generally.
How to Use Variables in Pattern Rules
Here's a step-by-step approach to writing a pattern rule using a variable:
- Identify the Pattern: Observe how the sequence changes from one term to the next. Is it increasing or decreasing? By a constant amount?
- Determine the Starting Point: Find the first term in the sequence that will serve as a reference point.
- Find the Consistent Change: Figure out the operation (addition, subtraction, multiplication, division) and the value involved to get from one term to the next.
- Introduce a Variable: Typically, the variable n is used to represent the term number (e.g., n=1 for the first term, n=2 for the second term, etc.).
- Write the Rule: Combine the starting point, the consistent change, and the variable to create the algebraic expression.
Examples Using Variables
- Example 1: Arithmetic Sequence
- Sequence: 7, 8, 9, 10, 11...
- Pattern: Add 1 to each previous term.
- Starting point: 7.
- Consistent change: +1 for each step.
- Rule (using n as the term number): 7 + (n - 1). This formula means you always start with 7 and then add 1 less than the term number. When n=1, the answer is 7. When n=2, the answer is 8, etc.
- Example 2: Another Arithmetic Sequence
- Sequence: 2, 4, 6, 8, 10
- Pattern: Add 2 to each previous term
- Starting Point: 2
- Consistent Change: +2 for each step
- Rule (using n as term number): *2n**. If you substitute term number 1 in, you get 2. If you substitute term number 2 in, you get 4, and so on.
- Example 3: Sequence with multiplication and addition
- Sequence: 5, 11, 17, 23, 29, ...
- Pattern: Add 6 to each previous term
- Starting Point: 5
- Consistent change: +6 for each step
- Rule (using n as the term number): *6 n - 1**. You multiply each term number by 6 and subtract 1 to get the correct number for each place in the sequence.
Practical Application
Pattern rules are beneficial because:
- Generalization: They provide a general formula to find any term in the sequence without having to calculate each preceding one.
- Problem Solving: They are helpful for solving problems where you need to predict future terms or find missing elements.
- Efficiency: They are more efficient for describing patterns rather than listing out all the terms.
By utilizing variables, pattern rules become powerful tools for analyzing and understanding sequences and numerical relationships.
