Represent pattern rules using algebraic expressions

A pattern rule is represented by an algebraic expression, such as $3n+2$ , in which the variable $n$ denotes the term position and the expression gives the corresponding term value. The relationship is interpreted through tables, sequences, or visual structures, with equivalent expressions recognized as describing the same rule; the focus is on one-variable additive or linear patterns with whole-number positions, not quadratic, exponential, multivariable, or more abstract generalizations.

Detailed Explanation: Represent pattern rules using algebraic expressions

A pattern rule uses a variable, usually $n$ , to show the term position. The expression tells you the value at position $n$ .

Example

The first terms of a pattern are:

5,\ 8,\ 11,\ 14,\ldots

Find an algebraic expression for the term at position $n$ .

Find how the pattern changes.

Each term increases by $3$ , so the rule includes $3n$ .
Compare $3n$ with the pattern.

If the rule were just $3n$ , the terms would be:

\begin{array}{c|c|c} \text{Position }n & 3n & \text{Pattern value}\\ \hline 1 & 3 & 5\\ 2 & 6 & 8\\ 3 & 9 & 11 \end{array}

Each pattern value is $2$ more than $3n$ .

Write the rule.

Therefore, the algebraic expression is

\boxed{3n+2}

Check the rule.

Substitute each position into $3n+2$ :

\begin{aligned} n=1 &: \quad 3(1)+2=5\\ n=2 &: \quad 3(2)+2=8\\ n=3 &: \quad 3(3)+2=11 \end{aligned}

The expression $3n+2$ correctly gives the value of every term in the pattern.

Learn by doing: Represent pattern rules using algebraic expressions

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Patterning - Equation from Rule for Increasing Arithmetic Pattern

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