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🔢 Finding the General Rule

Crack the code of any number pattern. Learn to spot the step, build a general rule like 3n + 1, and predict any term you like.

lessons

~15 min

to learn

🔢 Math

subject

Ages 6–12

level

Start the course →

What you’ll learn

Spot the StepLine up terms with their positions and find the common difference (the step) in a linear sequence.Organising a pattern by position (1st, 2nd, 3rd…) makes its step easy to spot. The steady jump between terms is the common difference, and it is the number you will multiply by when building the general rule. For 4, 7, 10, 13 the common difference is 3.
From Step to RuleBuild a general (nth-term) rule by multiplying the position by the common difference, then adjusting to fit the first term.To find the general rule, start with the common difference times n. That gives a multiples pattern; compare it to the real sequence and add or subtract to make it fit. For 4, 7, 10, 13 the step is 3, so 3n gives 3, 6, 9, 12 — one too small — so the rule is 3n + 1.
Test and Use the RuleUse a general rule to find any term instantly, and verify a rule by testing several positions.A general rule lets you find any term without counting up: for 3n + 1, the 10th term is 3 × 10 + 1 = 31. Always test a rule first by substituting the early positions — if 3n + 1 gives 4, 7, 10, 13 for n = 1, 2, 3, 4, the rule is trustworthy.

Questions this course answers

In the pattern 4, 7, 10, 13, what is the common difference?

Each term is 3 more than the one before, so the common difference is 3.

Why do we line the terms up with positions 1, 2, 3, 4?

Ordering terms by position makes the step between them clear and links each term to its place.

The step in a pattern is 5 and the first term is 5. What is the general rule?

Multiply the position by the step: 5n. Since 5 × 1 = 5 already matches the first term, no adjustment is needed.

Which general rule runs the pattern 5, 8, 11, 14?

The step is 3, so start with 3n (3, 6, 9, 12). Each real term is 2 more, so add 2: 3n + 2.

Using the rule 3n + 1, what is the 5th term?

Put n = 5 into the rule: 3 × 5 + 1 = 15 + 1 = 16.

What is the best way to check a general rule is correct?

A correct rule produces the right term for every position, so testing several positions confirms it.

Grounded in trusted sources

BBC Bitesize — sequences
NRICH — number patterns
Khan Academy — arithmetic sequences

Every Wunder lesson is built from real, reputable sources — never invented.

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