Find the first five terms of the sequence in which the nth term is given by

1. (i) t_n = 3n − 4,
2. (ii) t_n = 2 − 5n, and
3. (iii) t_n = n² − 2n + 3 for n ≥ 1.

Step-by-step solution

Idea: Substitute n = 1, 2, 3, 4, 5 into each rule. Take care with signs and with squaring.

Check: (i) goes up by 3 and (ii) goes down by 5, matching the 3n and −5n in the rules ✓. (iii) differences 1, 3, 5, 7 are the odd numbers, as expected for n² − 2n + 3 = (n − 1)² + 2 ✓.

Answer to write in the exam

Common mistakes that cost marks

• In (ii), subtracting before multiplying: 2 − 5 × 2 worked as (2 − 5) × 2 = −6. Multiply first: 2 − 10 = −8.
• In (iii), writing n² as 2n, so t₃ = 6 − 6 + 3 = 3. Square n: 3² = 9.
• In (i), writing the first term as 1 instead of −1 (3 − 4 = −1).

How this can come in the exam

Try one yourself

Find the first five terms of t_n = 2n² − 1.

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