| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Find term or common difference |
| Difficulty | Moderate -0.3 This is a straightforward two-part question testing standard arithmetic series formulas and basic algebraic manipulation. Part (a) requires direct application of S_n = n/2[2a + (n-1)d] and simple inequality solving. Part (b) involves solving a quadratic equation to find k and then substituting—all routine C2 techniques with no novel problem-solving required. Slightly easier than average due to its mechanical nature. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks |
|---|---|
| \(\frac{20}{7}[2a + (19 \times 7)] = 530\) | M1 |
| \(2a + 133 = 53, a = -40\) | M1 A1 |
| Answer | Marks |
|---|---|
| \(= -40 + 7k = -40 + 42 = 2\) | M1 A1 |
| Answer | Marks |
|---|---|
| \(u_1 = (1 + k)^2, u_2 = (2 + k)^2\) | B1 |
| Answer | Marks |
|---|---|
| \(k^2 = 2\) | M1 |
| \(k > 0 \therefore k = \sqrt{2}\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(u_3 = (3 + \sqrt{2})^2 = 9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2}\) | M1 A1 | (11) |
## (a)
### (i)
$\frac{20}{7}[2a + (19 \times 7)] = 530$ | M1 |
$2a + 133 = 53, a = -40$ | M1 A1 |
### (ii)
$= -40 + 7k = -40 + 42 = 2$ | M1 A1 |
## (b)
### (i)
$u_1 = (1 + k)^2, u_2 = (2 + k)^2$ | B1 |
$(2 + k)^2 = 2(1 + k)^2$
$4 + 4k + k^2 = 2 + 4k + 2k^2$
$k^2 = 2$ | M1 |
$k > 0 \therefore k = \sqrt{2}$ | M1 A1 |
### (ii)
$u_3 = (3 + \sqrt{2})^2 = 9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2}$ | M1 A1 | (11)
---\begin{enumerate}[label=(\alph*)]
\item An arithmetic series has a common difference of 7.
Given that the sum of the first 20 terms of the series is 530, find
\begin{enumerate}[label=(\roman*)]
\item the first term of the series, [3]
\item the smallest positive term of the series. [2]
\end{enumerate}
\item The terms of a sequence are given by
$$u_n = (n + k)^2, \quad n \geq 1,$$
where $k$ is a positive constant.
Given that $u_2 = 2u_1$,
\begin{enumerate}[label=(\roman*)]
\item find the value of $k$, [4]
\item show that $u_3 = 11 + 6\sqrt{2}$. [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q8 [11]}}