Can you write t₅, t₆, t₇ and t₈ for the sequence of triangular numbers?

Step-by-step solution

Idea: t_n means the term in position n. The nth triangular number is 1 + 2 + … + n, so each term is the previous term plus n.

1. Triangular numbers: t₁ = 1, t₂ = 3, t₃ = 6, t₄ = 10.
2. t₅ = t₄ + 5 = 10 + 5 = 15; t₆ = 15 + 6 = 21.1 mark
3. t₇ = 21 + 7 = 28; t₈ = 28 + 8 = 36.1 mark

Check: Using the formula n(n + 1)/2: t₈ = 8 × 9 ÷ 2 = 36 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Reading t₅ as ‘5 times t‘ or as the number 5. It means the 5th term of the sequence.
• Writing t₅ = 21 by counting from t₀. The first term 1 is t₁.
• Adding a fixed amount (e.g. +5 each time). The amount added equals the position number.

How this can come in the exam

Try one yourself

For the square numbers 1, 4, 9, 16, …, write t₆, t₉ and t₁₁.

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• Why is it useful to have an explicit formula for the nth term of a sequence?
• Using the explicit rule u_n = 2n − 1, find the 53rd term, the 108th term, and the 1170th term of the odd number sequence.
• Consider the sequence that is generated by the explicit formula s_n = 5n − 2.

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