| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: find specific terms |
| Difficulty | Moderate -0.8 Part (a) involves straightforward substitution into a simple recurrence relation and observing alternating behavior, while part (b) requires routine application of standard arithmetic progression formulas (nth term and sum). Both parts are mechanical applications of basic formulas with no problem-solving insight required, making this easier than a typical A-level question. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(u_2 = \frac{1}{2}\) | B1 | State \(\frac{1}{2}\). Allow \(0.5\) or \(\frac{2}{4}\) |
| \(u_3 = 4\) | B1 FT | State 4, following their \(u_2\). Follow through on their \(u_2\) (simplifying if possible). B0 for \(\frac{2}{0.5}\), \(\frac{7}{1/2}\) etc. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| periodic / alternating / repeating / oscillating / cyclic | B1 | Any correct description. Must be a mathematical term. Allow associated words e.g. 'repetitive'. B0 if additional incorrect terms (e.g. 'geometric') |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(a + 8d = 18\) | B1 | Allow any equivalent, including unsimplified. Must be correct when seen |
| \(\frac{9}{2}(2a + 8d) = 72\) | B1 | Allow any equivalent, including unsimplified. Must be correct when seen |
| \(a + 8d = 18\) and \(2a + 8d = 16\) | M1 | Attempt to solve simultaneously. M1 for eliminating a variable from two linear equations in \(a\) and \(d\). If balancing equations, must be intention to subtract (but allow \(a = 2\)). If substituting, allow sign errors (e.g. \(a = 8d - 18\)), but not operational errors |
| \(a = -2\), \(d = \frac{5}{2}\) | A1 | Obtain either \(a = -2\) or \(d = \frac{5}{2}\). Allow \(d = 2\frac{1}{2}\) or \(2.5\), but not unsimplified fractions |
| A1 | Obtain both \(a = -2\), \(d = \frac{5}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{9}{2}(a + 18) = 72\) | B1* | Allow any equivalent. Award B1 as soon as seen correct, even if subsequent error |
| \(a = -2\) | B1d* | Must come from correct equation |
| \(-2 + 8d = 18\) or \(\frac{9}{2}(-4 + 8d) = 72\) | M1 | Must be attempting either \(u_9 = 18\) or \(S_9 = 72\). Must be using correct formula |
| A1 FT | Obtain correct equation, following their \(a\). Allow any equivalent, including unsimplified | |
| \(d = \frac{5}{2}\) | A1 | Allow \(d = 2\frac{1}{2}\) or \(2.5\), but not unsimplified fractions |
## Question 5:
### Part (a)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $u_2 = \frac{1}{2}$ | B1 | State $\frac{1}{2}$. Allow $0.5$ or $\frac{2}{4}$ |
| $u_3 = 4$ | B1 FT | State 4, following their $u_2$. Follow through on their $u_2$ (simplifying if possible). B0 for $\frac{2}{0.5}$, $\frac{7}{1/2}$ etc. |
### Part (a)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| periodic / alternating / repeating / oscillating / cyclic | B1 | Any correct description. Must be a mathematical term. Allow associated words e.g. 'repetitive'. B0 if additional incorrect terms (e.g. 'geometric') |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $a + 8d = 18$ | B1 | Allow any equivalent, including unsimplified. Must be correct when seen |
| $\frac{9}{2}(2a + 8d) = 72$ | B1 | Allow any equivalent, including unsimplified. Must be correct when seen |
| $a + 8d = 18$ and $2a + 8d = 16$ | M1 | Attempt to solve simultaneously. M1 for eliminating a variable from two linear equations in $a$ and $d$. If balancing equations, must be intention to subtract (but allow $a = 2$). If substituting, allow sign errors (e.g. $a = 8d - 18$), but not operational errors |
| $a = -2$, $d = \frac{5}{2}$ | A1 | Obtain either $a = -2$ or $d = \frac{5}{2}$. Allow $d = 2\frac{1}{2}$ or $2.5$, but not unsimplified fractions |
| | A1 | Obtain both $a = -2$, $d = \frac{5}{2}$ |
**Alternative method using $\frac{n}{2}(a + l)$:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{9}{2}(a + 18) = 72$ | B1* | Allow any equivalent. Award B1 as soon as seen correct, even if subsequent error |
| $a = -2$ | B1d* | Must come from correct equation |
| $-2 + 8d = 18$ or $\frac{9}{2}(-4 + 8d) = 72$ | M1 | Must be attempting either $u_9 = 18$ or $S_9 = 72$. Must be using correct formula |
| | A1 FT | Obtain correct equation, following their $a$. Allow any equivalent, including unsimplified |
| $d = \frac{5}{2}$ | A1 | Allow $d = 2\frac{1}{2}$ or $2.5$, but not unsimplified fractions |
---5
\begin{enumerate}[label=(\alph*)]
\item A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by
$$u _ { 1 } = 4 \quad \text { and } \quad u _ { n + 1 } = \frac { 2 } { u _ { n } } \quad \text { for } n \geqslant 1 .$$
\begin{enumerate}[label=(\roman*)]
\item Write down the values of $u _ { 2 }$ and $u _ { 3 }$.
\item Describe the behaviour of the sequence.
\end{enumerate}\item In an arithmetic progression the ninth term is 18 and the sum of the first nine terms is 72. Find the first term and the common difference.
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2012 Q5 [8]}}