O-LevelAdditional MathematicsSeries4037/21 · May/June 2024 · Q810 Marks

In an arithmetic progression, the sum of the first 30 terms is -1065. The sum of the next 20 terms is -2210. Find the first term and the common difference.

(a)In an arithmetic progression, the sum of the first 30 terms is -1065. The sum of the next 20 terms is -2210. Find the first term and the common difference.[5]
(b)A geometric progression is such that the first term is 4 and the sum of the first three terms is 7. Find the two possible values of the common ratio and find the sum to infinity for the convergent progression.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[B1] S₃₀ = 30/2 {2a + 29d} = -1065
[B1] S₅₀ - S₃₀ = 50/2 {2a + 49d} - 30/2 {2a + 29d} = -2210 or S₅₀ = 50/2 {2a + 49d} = -2210 - 1065
[M1] Solves their linear equations in a and d as far as a = ... or d = ... (dep on an attempt to form their equations using at least one sum formula)
[A1] a = 8
[A1] d = -3
[B1] 4 + 4r + 4r² = 7 or 4(1-r³)/(1-r) = 7 oe
[B1] 4r² + 4r - 3 [= 0] oe
[M1] Solves or factorises their 3-term quadratic oe
[A1] r = 0.5, -1.5
[A1] 4 / (1-0.5) = 8 only, nfww

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