A-LevelMathematicsSeries9709/12 · May/June 2021 · Q88 Marks

The first, second and third terms of an arithmetic progression are a, 3/2a and b respectively, where a and b are positive constants. The first, second and third terms of a geometric progression are a, 18 and b + 3 respectively. (a) Find the values of a and b.

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

[B1] (a + b)/2 = 3/2 a ⇒ b = 2a
[B1] 18² = a(b + 3) OE or 2 correct statements about r from the GP, e.g. r = 18/a and b + 3 = 18r or r² = (b+3)/a
[M1] 324 = a(2a + 3) ⇒ 2a² + 3a – 324[= 0] or b² + 3b-648[= 0] or 6r² - r-12[= 0] or 4d² + 3d - 162[= 0]
[M1] (a – 12)(2a +27)[= 0] or (b-24)(b+27)[= 0] or (2r-3)(3r+4)[=0] or (d-6)(4d+27)[= 0]
[A1] a = 12, b = 24
[B1 FT] Common difference d = 6
[M1] S20 = 20/2 (2 × 12 + 19 × 6)
[A1] 1380

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Topic: Series, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source: A-Level Mathematics, May/June 2021 Paper 1 Variant 2, question 8.

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