Determine the AP whose third term is 16 and whose 7th term exceeds the 5th term by 12.

Step-by-step solution

Idea: ‘The 7th term exceeds the 5th term by 12’ means t₇ − t₅ = 12. Two steps of d separate them, so 2d = 12.

1. t₇ − t₅ = (a + 6d) − (a + 4d) = 2d = 12 ⇒ d = 6.1 mark
2. t₃ = a + 2d = 16 ⇒ a = 16 − 12 = 4.1 mark
3. The AP is 4, 10, 16, 22, 28, …1 mark

Check: 3rd term = 16 ✓; 7th term = 4 + 36 = 40, 5th term = 28, and 40 − 28 = 12 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Writing ‘exceeds by 12’ as t₇ = 12 × t₅ or t₅ − t₇ = 12.
• Taking 4d = 12 by subtracting 7 − 5 wrongly or using 7 and 5 as numbers of differences (it is 6d − 4d).
• Giving only a and d. ‘Determine the AP’ means write out its terms.

How this can come in the exam

Try one yourself

Determine the AP whose 2nd term is 11 and whose 10th term exceeds the 6th term by 20.

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