| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Sequence defined by formula |
| Difficulty | Standard +0.3 This is a straightforward multi-part question on arithmetic sequences requiring direct substitution, standard sum formula application, and basic GP identification. Part (iii) requires solving a simple equation using GP properties, and part (iv) is routine sum to infinity. Slightly above average due to the GP connection in parts (iii-iv), but all techniques are standard C2 material with no novel insight required. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04h 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 |
|---|---|---|
| Step | Mark | Guidance |
| State 80 | B1 | Just a list of numbers is fine, no need for labels |
| State 75 and 70 | B1 [2] | Ignore extra terms beyond \(u_3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Step | Mark | Guidance |
| Show intention to sum 1st 20 terms of an arithmetic sequence | M1 | Any recognisable attempt at the sum of an AP, including manual addition of terms – no need to list all terms, but intention (inc no. of terms) must be clear |
| Attempt use of correct sum formula for an AP, with \(n = 20\), \(a = 80\), \(d = \pm 5\) | M1 | Must use correct formula – only exception is \(10(2a + 9d)\). If using \(\frac{1}{2}n(a+l)\), must be a valid attempt at \(l\), either from \(a + 19d\) or from \(u_{20}\) |
| Obtain 650 | A1 [3] | Answer only gets full marks, as does manual addition |
| Answer | Marks | Guidance |
|---|---|---|
| Step | Mark | Guidance |
| Attempt to find \(u_p\) | M1* | Allow any valid method, inc informal. Allow if first and/or second terms of their GP are incorrect. Allow ratio of \(\frac{4}{3}\) if used correctly to find 3rd term \((60 \div \frac{4}{3})\) |
| Obtain 45 | A1 | Seen or implied. SR: M1* A0 if 45 results from using \(u_n = ar^n\). The following M1A1 are still available |
| Attempt to solve \(85 - 5p = k\) | M1d* | \(k\) must be from attempt at third term of GP. LHS could be \(80 + (p-1)(-5)\), from \(p^{\text{th}}\) term of the AP, but M0 if incorrect e.g. \(80 + (p-1)(5)\) |
| Obtain \(p = 8\) | A1 [4] | Allow full credit for answer only. Any variable, including \(n\) |
| Answer | Marks | Guidance |
|---|---|---|
| Step | Mark | Guidance |
| Use correct formula for sum to infinity | M1 | Must be from attempt at \(r\) for their GP |
| Obtain 320 | A1 [2] | A0 for 'tends to 320', 'approximately 320' etc |
## Question 6(i):
**Answer:** $u_1 = 80$, $u_2 = 75$, $u_3 = 70$
| Step | Mark | Guidance |
|------|------|----------|
| State 80 | B1 | Just a list of numbers is fine, no need for labels |
| State 75 and 70 | B1 [2] | Ignore extra terms beyond $u_3$ |
---
## Question 6(ii):
**Answer:** $S_{20} = \frac{20}{2}(2 \times 80 + 19 \times -5) = 650$
| Step | Mark | Guidance |
|------|------|----------|
| Show intention to sum 1st 20 terms of an arithmetic sequence | M1 | Any recognisable attempt at the sum of an AP, including manual addition of terms – no need to list all terms, but intention (inc no. of terms) must be clear |
| Attempt use of correct sum formula for an AP, with $n = 20$, $a = 80$, $d = \pm 5$ | M1 | Must use correct formula – only exception is $10(2a + 9d)$. If using $\frac{1}{2}n(a+l)$, must be a valid attempt at $l$, either from $a + 19d$ or from $u_{20}$ |
| Obtain 650 | A1 [3] | Answer only gets full marks, as does manual addition |
---
## Question 6(iii):
**Answer:** $r = \frac{60}{80} = 0.75$, $u_p = 80 \times 0.75^2 = 45$, $85 - 5p = 45$, $p = 8$
| Step | Mark | Guidance |
|------|------|----------|
| Attempt to find $u_p$ | M1* | Allow any valid method, inc informal. Allow if first and/or second terms of their GP are incorrect. Allow ratio of $\frac{4}{3}$ if used correctly to find 3rd term $(60 \div \frac{4}{3})$ |
| Obtain 45 | A1 | Seen or implied. **SR: M1* A0** if 45 results from using $u_n = ar^n$. The following M1A1 are still available |
| Attempt to solve $85 - 5p = k$ | M1d* | $k$ must be from attempt at third term of GP. LHS could be $80 + (p-1)(-5)$, from $p^{\text{th}}$ term of the AP, but M0 if incorrect e.g. $80 + (p-1)(5)$ |
| Obtain $p = 8$ | A1 [4] | Allow full credit for answer only. Any variable, including $n$ |
---
## Question 6(iv):
**Answer:** $S_\infty = \frac{80}{1-0.75} = 320$
| Step | Mark | Guidance |
|------|------|----------|
| Use correct formula for sum to infinity | M1 | Must be from attempt at $r$ for their GP |
| Obtain 320 | A1 [2] | A0 for 'tends to 320', 'approximately 320' etc |
---6 A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { n } = 85 - 5 n$ for $n \geqslant 1$.\\
(i) Write down the values of $u _ { 1 } , u _ { 2 }$ and $u _ { 3 }$.\\
(ii) Find $\sum _ { n = 1 } ^ { 20 } u _ { n }$.\\
(iii) Given that $u _ { 1 } , u _ { 5 }$ and $u _ { p }$ are, respectively, the first, second and third terms of a geometric progression, find the value of $p$.\\
(iv) Find the sum to infinity of the geometric progression in part (iii).
\hfill \mbox{\textit{OCR C2 2012 Q6 [11]}}