Skill: Describe pattern rules

A pattern rule tells how numbers in a pattern are made. Look for what changes from one number to the next.

• If the same amount is added or subtracted each time, it is an additive pattern.
• If each number is multiplied by the same amount, it is a multiplicative pattern.
• For a table, compare the input and output numbers to find the operation.

Worked example

Find the rule for this pattern and find the value at position $6$.

Position	$1$	$2$	$3$	$4$
Value	$5$	$8$	$11$	$14$

Step 1: Compare neighboring values.

$$8-5=3,\qquad 11-8=3,\qquad 14-11=3$$

The pattern adds $3$ each time.

So the term-to-term rule is:

Add $3$ to get the next value.

Step 2: Continue the pattern.

The next value is

$$14+3=17$$

The value after that is

$$17+3=20$$

So the value at position $6$ is $\boxed{20}$.

Step 3: Describe the rule using the position.

The value is also $3$ times the position, plus $2$:

This is a position rule. It tells how to find any value directly. For position $6$:

$$3(6)+2=20$$

Both rules describe the same pattern: the term-to-term rule says add $3$ each time, while the position rule says multiply the position by $3$, then add $2$.