| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Arithmetic progression with parameters |
| Difficulty | Moderate -0.3 This is a straightforward multi-part arithmetic sequence question requiring standard formula application. Parts (a) and (b) involve simple algebraic manipulation, (c) requires solving a quadratic equation, and (d)-(e) apply standard AP formulas. While it has multiple parts (11 marks total), each step is routine with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) \(= (t^2 - 5) - (t - 1) = t^2 - t - 4\) | M1 A1 | |
| (ii) \(= (t^2 - 5) + (t^2 - t - 4) = 2t^2 - t - 9\) | M1 A1 | |
| (b) \(2t^2 - t - 9 = 19\) | ||
| \(2t^2 - t - 28 = 0\) | ||
| \((2t+7)(t-4) = 0\) | M1 | |
| \(t > 0\) \(\therefore t = 4\) | A1 | |
| (c) \(a = 4 - 1 = 3\), \(d = 16 - 4 - 4 = 8\) | B1 | |
| \(u_{10} = 3 + (9 \times 8) = 3 + 72 = 75\) | M1 A1 | |
| (d) \(= \frac{40}{2}[6 + (39 \times 8)] = 20 \times 318 = 6360\) | M1 A1 | (11) |
**(a)** **(i)** $= (t^2 - 5) - (t - 1) = t^2 - t - 4$ | M1 A1 |
**(ii)** $= (t^2 - 5) + (t^2 - t - 4) = 2t^2 - t - 9$ | M1 A1 |
**(b)** $2t^2 - t - 9 = 19$ | |
$2t^2 - t - 28 = 0$ | |
$(2t+7)(t-4) = 0$ | M1 |
$t > 0$ $\therefore t = 4$ | A1 |
**(c)** $a = 4 - 1 = 3$, $d = 16 - 4 - 4 = 8$ | B1 |
$u_{10} = 3 + (9 \times 8) = 3 + 72 = 75$ | M1 A1 |
**(d)** $= \frac{40}{2}[6 + (39 \times 8)] = 20 \times 318 = 6360$ | M1 A1 | (11)The first two terms of an arithmetic series are $(t - 1)$ and $(t^2 - 5)$ respectively, where $t$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Find and simplify expressions in terms of $t$ for
\begin{enumerate}[label=(\roman*)]
\item the common difference of the series,
\item the third term of the series. [4]
\end{enumerate}
\end{enumerate}
Given also that the third term of the series is 19,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the value of $t$, [2]
\item show that the 10th term of the series is 75, [3]
\item find the sum of the first 40 terms of the series. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q9 [11]}}