Describing Patterns with Rules

Extending a pattern one step at a time is slow. A position-to-term rule lets you jump straight to any term using its position number $n$ ($n = 1$ for the first term, $2$ for the second, and so on).

For a pattern with a constant difference, the rule looks like

Worked example. For $4, 7, 10, 13, \ldots$, set up a table of position $n$ against term. The difference is $3$, so the rule starts with $3n$. Since $3 \times 1 = 3$ but the first term is $4$, add $1$: the rule is $\text{term} = 3n + 1$. Check $n = 4$: $3(4) + 1 = 13$. ✓

Exam checkpoint

A pattern rule should tell someone else how to make any term, not only the next one. Use a recursive rule for ‘what do I add or multiply each time?’ and a direct rule for ‘what is term $n$?’. Substitute a known term number to check that your rule really fits.

Short worked example. The sequence $4,7,10,13,\ldots$ has direct rule $3n+1$. For term 4: $3(4)+1=13$, which matches the table.