| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Sum of multiples or integers |
| Difficulty | Moderate -0.3 Part (a) is a standard proof by induction or formula derivation that appears in every C1 textbook. Part (b) requires applying the formula with some arithmetic manipulation to adjust ranges and identify arithmetic sequences, but follows a well-practiced pattern. This is slightly easier than average due to being a routine multi-part question with no novel problem-solving required. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae4.01a Mathematical induction: construct proofs |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(S_n = 1 + 2 + 3 + \ldots + n\) | B1 | |
| \(S_n = n + (n - 1) + (n - 2) + \ldots + 1\) | M1 | |
| adding, \(2S_n = n(n + 1)\) | M1 | |
| \(S_n = \frac{1}{2}n(n + 1)\) | A1 | |
| (b) (i) \(= S_{200} - S_{99}\) | M1 | |
| \(= \frac{1}{2} \times 200 \times 201 - \frac{1}{2} \times 99 \times 100\) | M1 | |
| \(= 20100 - 4950 = 15150\) | A1 | |
| (iii) \(= 3 \times 15150 = 45450\) | M1 A1 | (9 marks) |
**(a)** $S_n = 1 + 2 + 3 + \ldots + n$ | B1 |
$S_n = n + (n - 1) + (n - 2) + \ldots + 1$ | M1 |
adding, $2S_n = n(n + 1)$ | M1 |
$S_n = \frac{1}{2}n(n + 1)$ | A1 |
**(b)** (i) $= S_{200} - S_{99}$ | M1 |
$= \frac{1}{2} \times 200 \times 201 - \frac{1}{2} \times 99 \times 100$ | M1 |
$= 20100 - 4950 = 15150$ | A1 |
(iii) $= 3 \times 15150 = 45450$ | M1 A1 | (9 marks)\begin{enumerate}[label=(\alph*)]
\item Prove that the sum of the first $n$ positive integers is given by
$$\frac{1}{2}n(n + 1).$$ [4]
\item Hence, find the sum of
\begin{enumerate}[label=(\roman*)]
\item the integers from 100 to 200 inclusive,
\item the integers between 300 to 600 inclusive which are divisible by 3.
\end{enumerate} [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q8 [9]}}