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.

Step-by-step solution

Idea: Each term is the previous term multiplied by one less than itself. Start from 3 and apply the rule three times.

1. s₂ = s₁(s₁ − 1) = 3 × (3 − 1) = 3 × 2 = 6.½ mark
2. s₃ = s₂(s₂ − 1) = 6 × (6 − 1) = 6 × 5 = 30.½ mark
3. s₄ = s₃(s₃ − 1) = 30 × (30 − 1) = 30 × 29 = 870.½ mark
4. The first four terms are 3, 6, 30, 870.½ mark

Check: 30 × 29 = 30 × 30 − 30 = 900 − 30 = 870 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Reading s_n−1(s_n−1 − 1) as s_n−1 − 1 and getting 3, 2, 1, …. The bracket is multiplied by the term in front.
• Using the position number instead of the term: 2 × (2 − 1) = 2 for the second term. Use the previous term, 3.
• Arithmetic slip in 30 × 29 (e.g. 840). Use 30 × 30 − 30 = 870.

How this can come in the exam

Try one yourself

Find the first four terms of s₁ = 4, s_n = s_n−1(s_n−1 − 3) for n ≥ 2.

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