FP1 January 2013 Q8

EdexcelOld spec11 marksInduction Series

(a) Prove by induction that, for \(n \in \mathbb{Z}^+\), \[\sum_{r=1}^{n} r(r + 3) = \tfrac{1}{3}n(n + 1)(n + 5)\] (6)

(b) A sequence of positive integers is defined by \[\begin{aligned} u_1 &= 1, \\ u_{n+1} &= u_n + n(3n + 1), \qquad n \in \mathbb{Z}^+ \end{aligned}\] Prove by induction that \[u_n = n^2(n - 1) + 1, \qquad n \in \mathbb{Z}^+\] (5)

Mark scheme (a)

Scheme	Marks
If \(n = 1\), \(\displaystyle\sum_{r=1}^{n} r(r + 3) = 1 \times 4 = 4\) and \(\dfrac{1}{3}n(n + 1)(n + 5) = \dfrac{1}{3} \times 1 \times 2 \times 6 = 4\),	B1
(so true for \(n = 1\). Assume true for \(n = k\)) So \(\displaystyle\sum_{r=1}^{k+1} r(r + 3) = \dfrac{1}{3}k(k + 1)(k + 5) + (k + 1)(k + 4)\)	M1
\(= \dfrac{1}{3}(k + 1)\left[k(k + 5) + 3(k + 4)\right] = \dfrac{1}{3}(k + 1)\left[k^2 + 8k + 12\right]\)	A1
\(= \dfrac{1}{3}(k + 1)(k + 2)(k + 6)\) which implies is true for \(n = k + 1\)	dA1
As result is true for \(n = 1\) this implies true for all positive integers and so result is true by induction	dM1A1cso
	(6)

Notes

(a) First B for LHS=4 and RHS =4

First M for attempt to use \(\displaystyle\sum_{1}^{k} r(r + 3) + u_{k+1}\)

First A for \(\dfrac{1}{3}(k + 1)\), \(\dfrac{1}{3}(k + 2)\) or \(\dfrac{1}{3}(k + 6)\) as a factor before the final line

Second A dependent on first for \(\dfrac{1}{3}(k + 1)(k + 2)(k + 6)\) with no errors seen

Second M dependent on first M and for any 3 of ‘true for \(n = 1\)’ ‘assume true for \(n = k\)’ ‘true for \(n = k + 1\)’, ‘true for all \(n\)’ (or ‘true for all positive integers’) seen anywhere

Third A for correct solution only with all statements and no errors

Mark scheme (b)

Scheme	Marks
\(u_1 = 1^2(1 - 1) + 1 = 1\)	B1
(so true for \(n = 1\). Assume true for \(n = k\)) \(u_{k+1} = k^2(k - 1) + 1 + k(3k + 1)\)
\(= k(k^2 - k + 3k + 1) + 1 = k(k + 1)^2 + 1\) which implies is true for \(n = k + 1\)	M1, A1
As result is true for \(n = 1\) this implies true for all positive integers and so result is true by induction	M1A1cso
	(5)
	[11]

Notes

(b) First B for both some working and 1.

First M for \(u_{k+1} = u_k + k(3k + 1)\) and attempt to substitute for \(u_k\)

First A for \(k(k + 1)^2 + 1\) with some correct intermediate working and no errors seen

Second A for correct solution only with all statements and no errors