Higher June 2025 Paper 4 Q18
18
(a) A sequence is defined by
\(u_{n+1} = ku_n - 2\), \(u_2 = 7\) and \(u_3 = 29.5\).
Find the value of \(k\). [3]
(b) Another sequence is defined by \(u_{n+1} = 6u_n - 7\).
(i) Find the value of \(u_1\) in the case when \(u_2 = 2 \times u_1\). [2]
(ii) Explain what happens to the sequence \(u_{n+1} = 6u_n - 7\) when \(u_1 = 1.4\). [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 4.5 or \(4\frac{1}{2}\) or \(\frac{9}{2}\) oe nfww | 3 | M2 for \(\frac{29.5 + 2}{7}\) or M1 for \(29.5 = k \times 7 - 2\) oe Trials : M1 for each correct trial with value of \(k\) in \(k \times 7 - 2\) with the result to a maximum of 2 marks | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) 1.75 or \(1\frac{3}{4}\) or \(\frac{7}{4}\) oe | 2 | M1 for \(2u_1 = 6u_1 - 7\) oe or better Trials : M1 for one correct trial with input and output for \(u_1\) | accept any letter for \(u_1\) |
| (ii) Always get 1.4 oe | 1 | see appendix | |
Appendix: exemplar responses for Q18(b)(ii)
| Response | Mark |
|---|---|
| All the terms in the sequence will stay the same | 1 |
| Every term stays the same | 1 |
| The value never changes | 1 |
| The sequence repeats 1.4 | 1 |
| It has a period/order of 1 | 1 |
| \(u_n = 1.4\) | 1 |
| The sequence’s value stays the same | 1 (BOD) |
| It stays the same | 1 (BOD) |
| stays the same | 1 |
| \(u_2 = 1.4\) (needs \(u_n\)) | 0 |