| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Sum of specific range of terms |
| Difficulty | Moderate -0.3 This is a straightforward arithmetic series question requiring standard formula application. Part (a) uses S_n = n/2[2a + (n-1)d] with given values, part (b) sets up a simple equation from u_2 + u_3, then solving simultaneous equations and substituting. The sum from n=4 to 21 requires recognizing it as S_21 - S_3, which is routine manipulation. All techniques are standard C2 content with no novel insight required, making it slightly easier than average. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \([S_{21}] = \frac{21}{2}[2a + (21-1)d]\) | M1 |
| (a) | \(\frac{21}{2}[2a + 20d] = 168\) | m1 |
| (a) | \(21(a+10d) = 168 \Rightarrow a + 10d = 8\) | A1 |
| (b)(i) | \(a + d + a + 2d = 50 \quad (2a + 3d = 50)\) | M1 |
| (b)(i) | \(2(8 - 10d) + 3d = 50\) | m1 |
| (b)(i) | \(d = -2; \quad a = 28\) or \(12^{\text{th}}\) term = 8+d | A1 |
| (b)(i) | \((u_{12} =) 6\) | A1 |
| Total | 10 |
(a) | $[S_{21}] = \frac{21}{2}[2a + (21-1)d]$ | M1 | $\frac{21}{2}[2a + (21-1)d]$ OE
(a) | $\frac{21}{2}[2a + 20d] = 168$ | m1 | Forming correct eqn
(a) | $21(a+10d) = 168 \Rightarrow a + 10d = 8$ | A1 | 3 marks | AG $a + 10d = 8$ convincingly obtained with intermediate step shown eg 21(2a+20d)=168×2; 2a+20d=8×2
(b)(i) | $a + d + a + 2d = 50 \quad (2a + 3d = 50)$ | M1 | $a + d + a + 2d = 50$ OE in terms of a and d
(b)(i) | $2(8 - 10d) + 3d = 50$ | m1 | Solving $a + 10d = 8$ OE simultaneously with c's $2a + 3d = 50$ OE as far as correctly eliminating either a or d. PI by correct values for both d and either a or $u_{12}$.
(b)(i) | $d = -2; \quad a = 28$ or $12^{\text{th}}$ term = 8+d | A1 | $d = -2$ and either $a = 28$ or 8+d seen or used in part (b)(i).
(b)(i) | $(u_{12} =) 6$ | A1 | 4 marks | NMS scores 4/4 unless FIW
| **Total** | **10** |
---4 An arithmetic series has first term $a$ and common difference $d$.\\
The sum of the first 21 terms is 168 .
\begin{enumerate}[label=(\alph*)]
\item Show that $a + 10 d = 8$.
\item The sum of the second term and the third term is 50 .
The $n$th term of the series is $u _ { n }$.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $u _ { 12 }$.
\item Find the value of $\sum _ { n = 4 } ^ { 21 } u _ { n }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C2 2016 Q4 [10]}}