June 2024 Paper 2 Q7

OCR MEICurrent spec6 marksSequences & Series

7 A sequence is defined by the recurrence relation

\(u_{k+1} = u_k + 5\) with \(u_1 = -2\).

(a) Write down the values of \(u_2\), \(u_3\), and \(u_4\). [1]

(b) Explain whether this sequence is divergent or convergent. [1]

(d) Determine the value of \(\sum_{k=1}^{30} u_k\). [2]

Scheme	Marks	AO
\(u_2 = 3, u_3 = 8, u_4 = 13\)	B1	1.1
	[1]

Mark scheme (b)

Scheme	Marks	AO
divergent because difference between consecutive terms is not decreasing	E1	2.4
	[1]

Notes

E1: allow
divergent because ratio of consecutive terms is tending to 1 not 0;
divergent because the terms (in the sequence) are not tending to a finite limit oe
divergent because terms tend to infinity oe

do not allow
divergent because not convergent oe
divergent because terms (in the sequence) increase infinitely oe
divergent because terms get bigger oe

Mark scheme (c)

Scheme	Marks	AO
\(u_{30} = -2 + (30-1) \times 5\) used oe	M1	2.1
143	A1	1.1
	[2]

Notes

M1: eg may see \(3 + (29-1) \times 5\); \(a\) must be \(u_0, u_1, u_2, u_3\) or \(u_4\) in the AP and \(d\) must be 5; allow correct full list of terms from 3 to 138 for M1
must see at least eg \(-2 + 29 \times 5\)

A1: if M0 allow SCB1 for 143 not fully supported

Mark scheme (d)

Scheme	Marks	AO
\(S_{30} = \frac{30}{2}(-2 + \textit{their}\ 143)\) oe or \(S_{30} = \frac{30}{2}(2 \times (-2) + (30-1) \times 5)\) oe	M1	2.1
2115	A1	1.1
	[2]

Notes

M1: \(a\) must be \(-2\) and \(d\) must be 5; allow sum of full list of terms from \(-2\) to 143 for M1; allow if correct full list seen in part (c) only
must see at least \(15 \times (-2 + \text{their } 143)\) or \(15 \times (-4 + 29 \times 5)\) for M1

A1: if M0 allow SCB1 for 2115 not fully supported