Visual sequence proofs
Objective. Connect a growing diagram to an algebraic rule.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A visual sequence proof links a growing pattern of shapes to an algebraic rule by counting how many pieces are added at each step. Identify the constant part that stays the same and the part that grows with the pattern number n, then combine them into a formula. If a pattern starts with 3 squares and adds 2 more each time, the total at step n is 2n + 1, matching 3, 5, 7, 9, … Always check the formula against the actual diagram for at least two pattern numbers.
Worked example
A pattern of matchstick triangles has 3 matches for 1 triangle, 5 for 2 triangles, and 7 for 3 triangles. Find the rule for the number of matches for n triangles.
- 1Find the difference between terms: 5 − 3 = 2
- 2Start with 2n as the base term
- 3Substitute n = 1: 2(1) = 2, but the actual value is 3, so add 1
- 4The nth term is 2n + 1
Answer: 2n + 1
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils often count only the shapes visible in one diagram and generalise incorrectly, missing that a repeated 'arm' contributes 2 per step rather than 1. Always separate the fixed centre from the repeating part before writing the rule.
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