| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Find term or common difference |
| Difficulty | Moderate -0.8 This is a straightforward two-part question testing basic arithmetic progression formulas. Part (i) involves solving two simultaneous equations using the nth term formula, which is routine. Part (ii) is a direct application of the sum formula with given values—no problem-solving insight required, just mechanical substitution and verification. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| \(a + 19d = 10\), \(a + 49d = 70\) | M1 | Attempt to find \(d\) from simultaneous equations involving \(a + (n-1)d\) or equiv method |
| \(30d = 60 \Rightarrow d = 2\) | A1 | Obtain \(d = 2\) |
| \(a + (19 \times 2) = 10\) or \(a + (49 \times 2) = 70\) | M1 | Attempt to find \(a\) from \(a + (n-1)d\) or equiv |
| \(a = -28\) | A1 | Obtain \(a = -28\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(S = \frac{29}{2}(2 \times -28 + (29-1) \times 2) = 0\) | M1 | For relevant use of \(\frac{1}{2}n(2a + (n-1)d)\) |
| A1 | For showing the given result correctly | |
| AG |
**(i)**
$a + 19d = 10$, $a + 49d = 70$ | M1 | Attempt to find $d$ from simultaneous equations involving $a + (n-1)d$ or equiv method
$30d = 60 \Rightarrow d = 2$ | A1 | Obtain $d = 2$
$a + (19 \times 2) = 10$ or $a + (49 \times 2) = 70$ | M1 | Attempt to find $a$ from $a + (n-1)d$ or equiv
$a = -28$ | A1 | Obtain $a = -28$
**(ii)**
$S = \frac{29}{2}(2 \times -28 + (29-1) \times 2) = 0$ | M1 | For relevant use of $\frac{1}{2}n(2a + (n-1)d)$
| A1 | For showing the given result correctly
| | AG
---1 The 20th term of an arithmetic progression is 10 and the 50th term is 70 .\\
(i) Find the first term and the common difference.\\
(ii) Show that the sum of the first 29 terms is zero.
\hfill \mbox{\textit{OCR C2 2006 Q1 [6]}}