A-LevelMathematicsSeries9709/12 · May/June 2012 · Q78 Marks

In an arithmetic progression, the sum of the first n terms, denoted by Sn, is given by Sn = n² + 8n. Find the first term and the common difference.

(a)In an arithmetic progression, the sum of the first n terms, denoted by Sn, is given by Sn = n² + 8n. Find the first term and the common difference.[3]
(b)In a geometric progression, the second term is 9 less than the first term. The sum of the second and third terms is 30. Given that all the terms of the progression are positive, find the first term.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.

[B1] First term a = S₁ = 1² + 8(1) = 9; co.
[M1 A1] S₂ = 2² + 8(2) = 20. S₂ = a + (a+d) = 2a + d. 20 = 2(9) + d => d = 2; Realises that S₂ is a + (a + d). co (or equating n² + 8n with Sn and comparing coefficients).
[B1] Equation 1: ar = a - 9; co.
[B1] Equation 2: ar + ar² = 30; co.
[M1] Eliminates 'a' to form a quadratic in 'r': e.g. 9r(1+r)/(1-r) = 30, leading to 3r² + 13r - 10 = 0; Complete elimination of r or a.
[A1] Correct quadratic, for example, 3r² + 13r - 10 = 0 (or 2a² - 57a + 81 = 0); Solutions r = 2/3 or r = -5.
[A1] First term a = 27 (must choose positive r = 2/3 as terms are positive); co (condone 27 or 1.5).

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