| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Real-world AP: find term or total |
| Difficulty | Moderate -0.3 This is a straightforward application of standard arithmetic and geometric sequence formulas. Part (i) requires direct substitution into nth term and sum formulas for arithmetic sequences with clearly given values. Part (ii) involves finding the common ratio first, then applying the geometric series sum formula. All steps are routine with no conceptual challenges or problem-solving insight required, making it slightly easier than average but still requiring careful calculation across multiple parts. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(100 + 239 \times 5 = £1295\) | M1, A1 | For relevant use of \(a + (n-1)d\). For correct value 1295 |
| (ii) \(\frac{1}{2} \times 240 \times (100 + 1295) = £167400\) | M1, A1 | For relevant use of \(\frac{1}{2}n(a+l)\) or equivalent. For correct value 167400 |
| Answer | Marks | Guidance |
|---|---|---|
| Hence total is \(\frac{100(1.01139^{240} - 1)}{1.01139 - 1} = £124359\) | B1, M1, A1, M1, A1 | For correct statement of \(100r^{239} = 1500\). Attempt to find \(r\). For correct value 1.01. For relevant use of GP sum formula. For correct value 124359 (3 s.f. or better) |
**(i)** $100 + 239 \times 5 = £1295$ | M1, A1 | For relevant use of $a + (n-1)d$. For correct value 1295 | 2 marks
**(ii)** $\frac{1}{2} \times 240 \times (100 + 1295) = £167400$ | M1, A1 | For relevant use of $\frac{1}{2}n(a+l)$ or equivalent. For correct value 167400 | 2 marks
**(ii)** $100r^{239} = 1500 \Rightarrow r = 1.01139\ldots$
Hence total is $\frac{100(1.01139^{240} - 1)}{1.01139 - 1} = £124359$ | B1, M1, A1, M1, A1 | For correct statement of $100r^{239} = 1500$. Attempt to find $r$. For correct value 1.01. For relevant use of GP sum formula. For correct value 124359 (3 s.f. or better) | 5 marks
**Total: 9 marks**
---6 (i) John aims to pay a certain amount of money each month into a pension fund. He plans to pay $\pounds 100$ in the first month, and then to increase the amount paid by $\pounds 5$ each month, i.e. paying $\pounds 105$ in the second month, $\pounds 110$ in the third month, etc.
If John continues making payments according to this plan for 240 months, calculate
\begin{enumerate}[label=(\alph*)]
\item how much he will pay in the final month,
\item how much he will pay altogether over the whole period.\\
(ii) Rachel also plans to pay money monthly into a pension fund over a period of 240 months, starting with $\pounds 100$ in the first month. Her monthly payments will form a geometric progression, and she will pay $\pounds 1500$ in the final month.
Calculate how much Rachel will pay altogether over the whole period.
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2006 Q6 [9]}}