| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Mixed arithmetic and geometric |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing basic definitions of arithmetic and geometric progressions. Parts (i) and (ii) require simple formula application (GP: ar^n, AP: a+nd), part (iii) involves solving a quadratic equation, and part (iv) uses the standard sum formula. All techniques are routine for C2 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(r = \frac{x}{2}\) | M1 | |
| \(\therefore u_1 = x \times \frac{x}{2} = \frac{1}{2}x^2\) | M1 A1 | |
| (ii) \(a = 2, a + 2d = x\) | M1 | |
| \(\therefore d = \frac{1}{2}(x - 2)\) | ||
| \(u_{11} = 2 + [10 \times \frac{1}{2}(x - 2)] = 5x - 8\) | M1 A1 | |
| (iii) \(\frac{1}{2}x^2 = 5x - 8\) | M1 | |
| \(x^2 - 10x + 16 = 0\) | M1 | |
| \((x - 2)(x - 8) = 0\) | ||
| \(x \neq 2 \therefore x = 8\) | A1 | |
| (iv) \(d = \frac{1}{2}(8 - 2) = 3\) | B1 | |
| \(S_{50} = \frac{50}{2}[4 + (49 \times 3)] = 3775\) | M1 A1 | (12) |
| Answer | Marks |
|---|---|
| Total | (72) |
**(i)** $r = \frac{x}{2}$ | M1 |
$\therefore u_1 = x \times \frac{x}{2} = \frac{1}{2}x^2$ | M1 A1 |
**(ii)** $a = 2, a + 2d = x$ | M1 |
$\therefore d = \frac{1}{2}(x - 2)$ | |
$u_{11} = 2 + [10 \times \frac{1}{2}(x - 2)] = 5x - 8$ | M1 A1 |
**(iii)** $\frac{1}{2}x^2 = 5x - 8$ | M1 |
$x^2 - 10x + 16 = 0$ | M1 |
$(x - 2)(x - 8) = 0$ | |
$x \neq 2 \therefore x = 8$ | A1 |
**(iv)** $d = \frac{1}{2}(8 - 2) = 3$ | B1 |
$S_{50} = \frac{50}{2}[4 + (49 \times 3)] = 3775$ | M1 A1 | (12)
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**Total** | (72) |9. The first two terms of a geometric progression are 2 and $x$ respectively, where $x \neq 2$.\\
(i) Find an expression for the third term in terms of $x$.
The first and third terms of arithmetic progression are 2 and $x$ respectively.\\
(ii) Find an expression for the 11th term in terms of $x$.
Given that the third term of the geometric progression and the 11th term of the arithmetic progression have the same value,\\
(iii) find the value of $x$,\\
(iv) find the sum of the first 50 terms of the arithmetic progression.
\hfill \mbox{\textit{OCR C2 Q9 [12]}}