| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Arithmetic progression with parameters |
| Difficulty | Moderate -0.8 This is a straightforward application of the arithmetic series formula S_n = n/2[2a + (n-1)d] with clear scaffolding. Part (a) is shown for students, parts (b-c) involve routine substitution and solving simultaneous equations, and part (d) requires finding a single term. The question requires only standard recall and basic algebraic manipulation with no problem-solving insight needed. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Uses \(\frac{n}{2}(2a+(n-1)d)\) with \(n=10\) to give \(10a + 45d = 395\) | B1* | Use correct sum formula with \(n=10\), \(S=395\); proceeds to given answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Uses \(\frac{n}{2}(2a+(n-1)d)\) with \(n=18\) and \(S=927\) | M1 | Obtain correct second equation |
| \(18a + 153d = 927\) or \(2a + 17d = 103\) | A1 | Simplified equation; sight of one of these scores both marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Solves simultaneous equations to find either \(a\) or \(d\) | M1 | Do not concern yourself with process |
| \(a = 26\) and \(d = 3\) | A1, A1 | A1: correct \(a\) or \(d\) (just one); A1: correct \(a\) and \(d\) (both) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Uses \(a + (n-1)d\) with \(n = 20\) | M1 | Uses correct formula for \(n\)th term with their \(a\) and \(d\) but with \(n=20\); look for \('a'+19'd'\) |
| \(= 83\) | A1 | Correct answer |
## Question 9:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Uses $\frac{n}{2}(2a+(n-1)d)$ with $n=10$ to give $10a + 45d = 395$ | B1* | Use correct sum formula with $n=10$, $S=395$; proceeds to given answer |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Uses $\frac{n}{2}(2a+(n-1)d)$ with $n=18$ and $S=927$ | M1 | Obtain correct second equation |
| $18a + 153d = 927$ or $2a + 17d = 103$ | A1 | Simplified equation; sight of one of these scores both marks |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Solves simultaneous equations to find either $a$ or $d$ | M1 | Do not concern yourself with process |
| $a = 26$ and $d = 3$ | A1, A1 | A1: correct $a$ or $d$ (just one); A1: correct $a$ and $d$ (both) |
### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Uses $a + (n-1)d$ with $n = 20$ | M1 | Uses correct formula for $n$th term with their $a$ and $d$ but with $n=20$; look for $'a'+19'd'$ |
| $= 83$ | A1 | Correct answer |\begin{enumerate}
\item In a large theatre there are 20 rows of seats.
\end{enumerate}
The number of seats in the first row is $a$, where $a$ is a constant.
In the second row the number of seats is $( a + d )$, where $d$ is a constant. In the third row the number of seats is $( a + 2 d )$, and on each subsequent row there are $d$ more seats than on the previous row. The number of seats in each row forms an arithmetic sequence.
The total number of seats in the first 10 rows is 395\\
(a) Use this information to show that $10 a + 45 d = 395$
The total number of seats in the first 18 rows is 927\\
(b) Use this information to write down a second simplified equation relating $a$ and $d$.\\
(c) Solve these equations to find the value of $a$ and the value of $d$.\\
(d) Find the number of seats in the 20th row of the theatre.
\hfill \mbox{\textit{Edexcel C12 2016 Q9 [8]}}