A-LevelFurther MathematicsSummation of series9231/12 · Oct/Nov 2022 · Q17 Marks

Use the list of formulae (MF19) to find `\sum_{r=1}^{n} r(r+2)` in terms of n, simplifying your answer.

(a)Use the list of formulae (MF19) to find `\sum_{r=1}^{n} r(r+2)` in terms of n, simplifying your answer.[2]
(b)Express `1/(r(r+2))` in partial fractions and hence find `Σᵣ₌₁ⁿ 1/(r(r+2))` in terms of n.[4]
(c)Deduce the value of `Σᵣ₌₁^(∞) 1/(r(r+2))`.[1]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[M1] `Σᵣ₌₁ⁿ r(r+2) = Σᵣ₌₁ⁿ r² + 2Σᵣ₌₁ⁿ r = 1/6n(n+1)(2n+1) + n(n+1)` (Uses List of formulae (MF19).)
[A1] `1/6n(n+1)(2n+7)` (At least all like terms collected and fractions simplified.)
[M1 A1] `1/(r(r+2)) = 1/2(1/r - 1/(r+2))` (Expresses in partial fractions.)
[M1] `Σᵣ₌₁ⁿ 1/(r(r+2)) = 1/2 ( (1 - 1/3) + (1/2 - 1/4) + (1/3 - 1/5) + … + (1/(n-1) - 1/(n+1)) + (1/n - 1/(n+2)) )` (Writes enough terms.)
[A1] `1/2 (3/2 - 1/(n+1) - 1/(n+2))` (OE)
[B1 FT] `3/4` (FT their part (b) of correct form.)

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Topic: Summation of series, A-Level Further Mathematics (syllabus 9231). Structured question, 7 marks.
Source: A-Level Further Mathematics, Oct/Nov 2022 Paper 1 Variant 2, question 1.

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