Can you find the rule describing the nth term of the sequence of square numbers?

Step-by-step solution

Idea: Write each term next to its position and look for the link between the two.

1. Position → term: 1 → 1, 2 → 4, 3 → 9, 4 → 16, 5 → 25.½ mark
2. Each term is its position multiplied by itself: 4 = 2 × 2, 9 = 3 × 3, 25 = 5 × 5.½ mark
3. So the explicit rule is t_n = n².1 mark

Check: t₆ = 36 and t₁₀ = 100, which are the 6th and 10th square numbers ✓.

Answer to write in the exam

Common mistakes that cost marks

• Writing 2n (doubling) instead of n² (squaring): 2 × 3 = 6, but the 3rd square number is 9.
• Writing n + 3 or n + 2 from the first two terms only. Test the rule on several terms.
• Using the difference rule (add 3, 5, 7, …) as the answer. That describes how terms grow, not the nth term.

How this can come in the exam

Try one yourself

Find the rule for the nth term of 0, 3, 8, 15, 24, …

More questions like this

• Here is the sequence of the first ten prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Do you see any pattern in this sequence? Can you think of a rule that can predict the next few prime numbers?
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• Consider the sequence 1, 4, 7, 10, 13, … . The nth term is t_n = 3n − 2 (verify this for yourself).
• Find the first four terms of the sequence given by the recursive rule u₁ = 1, u_n = 2u_n−1 + 3 for n ≥ 2. Is 133 a term of this sequence?
• Find the first four terms of the sequence given by the recursive rule s₁ = 3, s_n = s_n−1 (s_n−1 − 1) for n ≥ 2.

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