| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Prove sum formula |
| Difficulty | Moderate -0.8 Part (a) is a standard bookwork proof of the arithmetic series formula that appears in every C1 textbook. Parts (b) and (c) involve straightforward application of the formula to a context problem with simple simultaneous equations. This is routine practice material with no novel problem-solving required. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Leave blank |
| Answer | Marks | Guidance |
|---|---|---|
| \(S = a + (a+d) + \ldots + [a+(n-1)d]\) | B1 | |
| \(S = [a+(n-1)d] + \ldots + a\) | M1 | |
| Add: \(2S = n[2a+(n-1)d]\), \(\therefore S = \frac{1}{2}n[2a+(n-1)d]\) \((*)\) | M1 A1 | 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(a + 15d = 6\) | B1 | |
| \(\frac{1}{2}n[2a+(n-1)d] = 8(2a+15d) = 72\) | M1 A1 | |
| Solve simultaneously: \(a = 3\) | M1 A1 | 5 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(a=3\): \(15d = 6-3 = 3\), \(d = 0.2\) | M1 A1 | 2 marks |
## Question 9:
### Part (a):
$S = a + (a+d) + \ldots + [a+(n-1)d]$ | B1 |
$S = [a+(n-1)d] + \ldots + a$ | M1 |
Add: $2S = n[2a+(n-1)d]$, $\therefore S = \frac{1}{2}n[2a+(n-1)d]$ $(*)$ | M1 A1 | **4 marks**
### Part (b):
$a + 15d = 6$ | B1 |
$\frac{1}{2}n[2a+(n-1)d] = 8(2a+15d) = 72$ | M1 A1 |
Solve simultaneously: $a = 3$ | M1 A1 | **5 marks**
### Part (c):
$a=3$: $15d = 6-3 = 3$, $d = 0.2$ | M1 A1 | **2 marks**
**Total: 11 marks**
---9. An arithmetic series has first term $a$ and common difference $d$.
\begin{enumerate}[label=(\alph*)]
\item Prove that the sum of the first $n$ terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .\) \\
(4) \\
A polygon has 16 sides. The lengths of the sides of the polygon, starting with the shortest side, form an arithmetic sequence with common difference $d \mathrm {~cm}$. \\
The longest side of the polygon has length 6 cm and the perimeter of the polygon is 72 cm . \\
Find
\item the length of the shortest side of the polygon, \\
(5)
\item the value of $d$. \\
(2) $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ \\
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\hfill \mbox{\textit{Edexcel C1 Q9 [11]}}