A2 June 2024 Paper 2 Q5

AQACurrent spec3 marksSeries

5 The first four terms of the series \(S\) can be written as

\[S = (1 \times 2) + (2 \times 3) + (3 \times 4) + (4 \times 5) + \ldots\]

(a) Write an expression, using \(\sum\) notation, for the sum of the first \(n\) terms of \(S\) [1 mark]

(b) Show that the sum of the first \(n\) terms of \(S\) is equal to\[\frac{1}{3}n(n + 1)(n + 2)\] [2 marks]

Scheme	Marks	AO
Obtains \(\displaystyle\sum_{r=1}^{n} r(r + 1)\) OE ISW Condone \(\displaystyle\sum_{r=1}^{n} r^2 + r\)	B1	2.5
	(1)

Mark scheme (b)

Scheme	Marks	AO
Uses \(\dfrac{1}{6}n(n + 1)(2n + 1)\) and \(\dfrac{1}{2}n(n + 1)\)	M1	1.1a
Completes fully correct working, with at least one intermediate step, to obtain \(\dfrac{1}{3}n(n + 1)(n + 2)\) AG LHS of typical solution not required.	R1	2.1
	(2)
	(3 marks)

Typical solution

\[\begin{aligned} \sum_{r=1}^{n} (r^2 + r) &= \frac{1}{6}n(n + 1)(2n + 1) + \frac{1}{2}n(n + 1) \\ &= \frac{1}{6}n(n + 1)(2n + 1 + 3) \\ &= \frac{1}{3}n(n + 1)(n + 2) \end{aligned}\]