June 2025 Paper 1 Q3

OCR ACurrent spec6 marksSequences & Series

(a) A sequence has terms \(u_1, u_2, u_3, \ldots\) defined by \(u_1 = 3\) and \(u_{n+1} = u_n^{\,2} - 5\) for \(n \geqslant 1\).

(i) Find the values of \(u_2\), \(u_3\) and \(u_4\). [2]

(ii) Describe the behaviour of the sequence. [1]

(b) The second, third and fourth terms of a geometric progression are 12, 8 and \(\frac{16}{3}\).

Determine the sum to infinity of this geometric progression. [3]

Mark scheme (a)

Scheme	Marks	AO
(i) \(u_2 = 4\)	B1	1.1
\(u_3 = 11\), \(u_4 = 116\)	B1FT	1.1
	[2]
(ii) Increasing or diverging	B1	2.4
	[1]

Notes

(a)(i)

B1: No need to label terms as \(u_2\) etc, and condone 3 appearing at start of list ie 3, 4… would be B1

B1FT: Both correct. FT their \(u_2\). Ignore any further terms given

(a)(ii)

B1: B0 if additional incorrect statement such as ‘diverging AP’ or ‘increasing exponentially’

‘Increasing monotonically’ is fine, but B0 if just ‘monotonic’

Mark scheme (b)

Scheme	Marks	AO
\(r = \dfrac{2}{3}\) oe	M1	3.1a
\(a = 18\)	M1	2.1
\(S_\infty = \dfrac{18}{1-\frac{2}{3}} = 54\)	A1	1.1
	[3]

Notes

M1: Attempt to find \(r\). Attempts at \(a\) and \(r\) could be in either order, with M1 awarded for a correct method with a possibly incorrect \(a\) or \(r\)

M1: Attempt to find \(a\). Correct \(a\) and/or \(r\) seen in formula for sum to infinity would imply the M mark(s)

A1: Use correct \(a\) and \(r\) to obtain sum to infinity as 54

‘Determine’ so some method needed. Minimum is \(\dfrac{18}{\frac{1}{3}} = 54\).

SC B1 for 54 with no working shown.