| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2018 |
| Session | March |
| Marks | 10 |
| Topic | Arithmetic Sequences and Series |
| Type | Real-world AP: find n satisfying a condition |
| Difficulty | Moderate -0.8 This is a straightforward application of arithmetic sequences requiring only direct formula substitution (finding a₁₀ and using sum formula). The real-world context and part (iii) add minimal difficulty since the mathematical content is purely routine AS-level arithmetic sequence work with no problem-solving insight required. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| \(u_{10} = 150 + 9 \times 16 = 294\) ice creams; profit \(= 294 \times £1.25 = £367.50\) | B1, M1, A1FT | Identify AP, with \(a = 150\) and \(d = 16\); Correct \(u_{10}\); Correct profit for their \(u_{10}\); Units required |
| Answer | Marks | Guidance |
|---|---|---|
| £5000 \(\div\) £1.25 = 4000; \(S_N = 0.5N(300 + (N - 1)16)\); \(150N + 8N(N-1) > 4000\); \(8N^2 + 142N - 4000 > 0\); \(N = 15.18\) (and possibly -32.9); Week 16 | B1, M1, A1, M1, A1 | Identify that 4000 sales are reqd; Attempt \(S_N\) of AP, with \(a = 150\) and \(d = 16\); Link to 4000 (any sign) and rearrange to 3 term quadratic; Attempt to solve quadratic; Conclude with Week 16 only; Or use \(d = £20\); Or \(d = 20\); Or link to 5000 (any sign) and rearrange to 3 term quadratic; BC; Allow 'during Week 16' |
| Answer | Marks | Guidance |
|---|---|---|
| Sales cannot continue to increase for ever; Weekly sales could fluctuate depending on the weather | E1, E1 | Refer to trend not continuing; Refer to changes week by week; Any two different reasons |
**(i)**
| $u_{10} = 150 + 9 \times 16 = 294$ ice creams; profit $= 294 \times £1.25 = £367.50$ | B1, M1, A1FT | Identify AP, with $a = 150$ and $d = 16$; Correct $u_{10}$; Correct profit for their $u_{10}$; Units required |
**(ii)**
| £5000 $\div$ £1.25 = 4000; $S_N = 0.5N(300 + (N - 1)16)$; $150N + 8N(N-1) > 4000$; $8N^2 + 142N - 4000 > 0$; $N = 15.18$ (and possibly -32.9); Week 16 | B1, M1, A1, M1, A1 | Identify that 4000 sales are reqd; Attempt $S_N$ of AP, with $a = 150$ and $d = 16$; Link to 4000 (any sign) and rearrange to 3 term quadratic; Attempt to solve quadratic; Conclude with Week 16 only; Or use $d = £20$; Or $d = 20$; Or link to 5000 (any sign) and rearrange to 3 term quadratic; BC; Allow 'during Week 16' |
**(iii)**
| Sales cannot continue to increase for ever; Weekly sales could fluctuate depending on the weather | E1, E1 | Refer to trend not continuing; Refer to changes week by week; Any two different reasons |5 An ice cream seller expects that the number of sales will increase by the same amount every week from May onwards. 150 ice creams are sold in Week 1 and 166 ice creams are sold in Week 2. The ice cream seller makes a profit of $\pounds 1.25$ for each ice cream sold.\\
(i) Find the expected profit in Week 10.\\
(ii) In which week will the total expected profits first exceed $\pounds 5000$ ?\\
(iii) Give two reasons why this model may not be appropriate.
\hfill \mbox{\textit{OCR H240/01 2018 Q5 [10]}}