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Learn Index · Patterns and Algebra Grade 3–5

Pattern rules

A statement that tells how to get from one term to the next, or from position to value.

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Describing a pattern

A pattern rule tells exactly how a pattern works so anyone can continue it. A good rule includes the starting value and how the pattern changes.

"Start at 4 and add 3 each time" → 4, 7, 10, 13, 16 …

Two kinds of rules

Type Says Example for 4, 7, 10, 13
Recursive rule How to get the next term from the one before Start at 4; add 3 each time
Explicit rule How to get any term from its term number Term = 3 × term number + 1

The recursive rule is easy to say but slow for far-away terms. To find the 100th term you’d have to add 3 ninety-nine times! The explicit rule jumps straight there: 3 × 100 + 1 = 301.

Finding an explicit rule

Use a table:

Term number (n) 1 2 3 4
Term 4 7 10 13
  1. Find the difference between terms: 3. That’s the multiplier.
  2. Multiply each term number by 3: 3, 6, 9, 12.
  3. Compare with the actual terms: each is 1 more.
  4. Rule: term = 3n + 1.

Check: n = 2 → 3 × 2 + 1 = 7 ✓.

Rules for shape patterns

Picture a row of squares made with toothpicks:

Squares 1 2 3 4
Toothpicks 4 7 10 13

The first square needs 4 toothpicks, and each new square shares a side, so it adds 3. Recursive: start at 4, add 3. Explicit: toothpicks = 3 × squares + 1.

Multiplying patterns

Not all rules add. "Start at 2 and multiply by 2 each time" gives 2, 4, 8, 16, 32.

Checking a rule

Always test your rule on at least three terms. If it fails one, adjust it.

Worked examples

Example 1: Write a recursive rule for 50, 45, 40, 35.

Start at 50 and subtract 5 each time.

Example 2: Find an explicit rule for 5, 9, 13, 17.

Difference 4 → 4n: 4, 8, 12, 16. Each term is 1 more. Term = 4n + 1.

Example 3: Use term = 2n + 3 to find the 20th term.

2 × 20 + 3 = 43.

Common mistakes

  • Leaving out the starting value in a recursive rule.
  • Using the difference alone as the explicit rule (4n instead of 4n + 1).
  • Testing only one term.

Try it yourself

  1. Write a recursive rule for 3, 8, 13, 18.
  2. Find an explicit rule for 6, 11, 16, 21.
  3. Using term = 5n − 2, what is the 10th term?
  4. A pattern starts at 1 and triples each time. Write the first four terms.

Answers: 1) Start at 3 and add 5 each time. 2) Term = 5n + 1 3) 48 4) 1, 3, 9, 27

Quick recap

  • A pattern rule gives the starting value and the change.
  • Recursive rules go term to term; explicit rules use the term number.
  • Test every rule on several terms.