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?

Step-by-step solution

Idea: A recursive rule makes each term from the one before: double the previous term and add 3. Keep going until you pass 133.

1. u₂ = 2u₁ + 3 = 2 × 1 + 3 = 5.½ mark
2. u₃ = 2u₂ + 3 = 2 × 5 + 3 = 13; u₄ = 2u₃ + 3 = 2 × 13 + 3 = 29.1 mark
3. First four terms: 1, 5, 13, 29. Next: u₅ = 2 × 29 + 3 = 61, u₆ = 2 × 61 + 3 = 125, u₇ = 2 × 125 + 3 = 253.1 mark
4. The terms keep increasing, and 125 < 133 < 253, so 133 is skipped. 133 is not a term.½ mark

Check: Each term is odd (double anything, add 3 → odd), and the list 1, 5, 13, 29, 61, 125, 253 has no 133 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Calculating 2 × 1 + 3 as 2 × 4 = 8. Multiply first, then add: 2 + 3 = 5.
• Using u₁ every time instead of the previous term (getting 5, 5, 5, …).
• Stopping at 29 and saying 133 is not a term without showing that the terms jump from 125 to 253.

How this can come in the exam

Try one yourself

Find the first five terms of u₁ = 3, u_n = 2u_n−1 − 1 for n ≥ 2. Is 65 a term?

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