Steps to Find Algebraic Form of Linear Pattern Rule
- Identify the Pattern:
- Examine the given sequence or pattern.
- Note the relationship between the position of each term and its value.
- Determine the Common Difference or Ratio:
- For linear patterns, determine the common difference between consecutive terms.
- If the pattern is increasing, note how much each term increases by.
- If the pattern is decreasing, note how much each term decreases by.
- Write the General Form:
- Start with the general form of a linear pattern, which is often expressed as \( a_n = mn + b \).
- Here, \( a_n \) is the nth term, \( m \) is the common difference, and \( b \) is the initial term.
- Substitute Known Values:
- Use the identified common difference and any known term values to substitute into the general form.
- This will give you an equation in terms of \( n \) (the position of the term).
- Simplify the Equation:
- Simplify the equation by combining like terms and performing any necessary arithmetic operations.
- Finalize the Algebraic Form:
- Write the simplified equation as the final algebraic form of the linear pattern rule.
- The equation should represent the nth term of the pattern based on the position \( n \).
Example:
Let's say you have the sequence 3, 7, 11, 15, 19, ...
- Identify the pattern: Each term increases by 4.
- Determine the common difference: \( m = 4 \).
- Write the general form: \( a_n = 4n + b \).
- Substitute known values: Using the first term, \( a_1 = 3 \), you get \( 3 = 4(1) + b \).
- Simplify the equation: Solve for \( b \) to find \( b = -1 \).
- Finalize the algebraic form: The linear pattern rule is \( a_n = 4n - 1 \).