L2 · The Pattern Rule: One Operation, in Words and in Symbols

Learning goal

“I can write the pattern rule that connects position to term for an arithmetic sequence - in words and as an algebraic expression - when the rule uses ONE operation: term = number x position (like 4n) or term = position + number (like n + 7), and I can use the rule to find a term at any position.”

Pattern rule — a statement that produces the term directly from the position, without counting through all the steps in between.

Variable — a letter (we use n) that stands for the position. n can be 1, 2, 3, ... - any position you choose.

Algebraic expression — a rule written with numbers, operations and a variable, like 4n or n + 7.

Coefficient form — products with a variable are written WITHOUT the x sign: 4n means 4 x n.

Worked example

Find the rule for 4, 8, 12, 16. Step 1: Table - Position: 1, 2, 3, 4 - Term: 4, 8, 12, 16. Constant difference +4. Step 2: Compare term to position: position 1 -> 4 = 1 x 4; position 2 -> 8 = 2 x 4; position 3 -> 12 = 3 x 4. Every term is position x 4. Step 3: In words: "Multiply the position by 4." In symbols: term = 4n. Step 4 (the payoff): Position 20 without counting: term = 4 x 20 = 80. The rule jumps straight to any position - that is what rules are FOR. Say it plainly: when the terms are exactly the multiples of the constant difference (the sequence starts AT the difference), the rule is "position x difference".

Check yourself — explain it

The rule for 7, 14, 21, 28 must be 7n and not n + 7, because...

Paired problem — example still visible

Rule for 6, 12, 18, 24 -> 6n; check position 3: 6 x 3 = 18

Faded completion — last step is yours

"Sequence 9, 18, 27, 36: rule is ___n; term at position 8 is 9 x ___ = ___" -> 9; 8; 72

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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