Patterns and Rules

Patterns follow rules. A rule for term n lets us find any term, decide which term a number is, and test whether a number belongs at all.

Key ideas

If a sequence goes up by a each time, its rule is term n = a × n + b, where b is the first term minus a.
To find which term a number is, undo the rule: take away b, then divide by a.
A number belongs to the sequence only if that division gives a whole number.
Some sequences mix rules: two sequences can take turns, or the steps can alternate (add, then double).
In growing patterns the gaps may grow too; square-number patterns use n × n.
A letter pattern can jump different amounts in turn; a letter code moves every letter the same number of places.

Remember

5, 8, 11, 14, …	term n = 3n + 2; term 20 = 62
Which term is 62?	(62 − 2) ÷ 3 = 20
Is 70 a term?	(70 − 2) ÷ 3 is not whole, so no
Tile pattern n × n + b	3, 6, 11, 18 (b = 2) → term 10 = 102
Each term is the sum of the two before	2, 3, 5, 8, 13, 21, …
Double and add 1	2, 5, 11, 23, 47, …

Watch out

Using the first term as b: in 5, 8, 11 the rule is 3n + 2, not 3n + 5.
Saying a number is a term without checking that the division comes out whole.
Finding one rule that fits the first two terms and not testing the rest.

Worked example

(a) A pattern uses 3, 5, 7, 9 sticks for terms 1, 2, 3, 4. How many sticks does term 17 use?
(b) A pattern uses 10, 16, 22, 28 sticks for terms 1, 2, 3, 4. Which is the first term to use more than 297 sticks?

Working: (a) term n uses 2 × n + 1 sticks: 2 × 17 + 1 = 35; (b) term n uses 6 × n + 4 sticks; term 48 uses 292, term 49 uses 298

Answer: (a) 35; (b) 49