Skill: 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 $n$.

1. Find how the pattern changes.

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

2. 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 $3n$.

3. Write the rule.

   Therefore, the algebraic expression is

$$\boxed{3n+2}$$

4. 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.