| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2014 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Real-world AP: find term or total |
| Difficulty | Easy -1.2 This is a straightforward arithmetic sequence problem requiring only basic formula application: finding a term using a=14000, d=1500; calculating sum of 9 terms; finding Anna's starting salary from the 10th term condition; and comparing two arithmetic series sums. All steps are routine with no problem-solving insight needed, making it easier than average A-level questions. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(14000 + 8 \times 1500 = 14000 + 12000 = £26000\) | M1, A1* | Uses \(S = a + (n-1)d\) with \(a=14000, d=1500\) and \(n=8, 9\) or \(10\) in an attempt to find salary in year 9. Accept a sequence written out only if all terms up to year 9 are included—Allow no errors. csa 26000. It is acceptable to write a sequence for both the 2 marks. FYI the terms are \(14000, 15500, 17000, 18500, 20000, 21500, 23000, 24500, 26000\) |
| (2 marks) | ||
| (b) \(S_n = \frac{n}{2}(a+l) = \frac{9}{2} \times (14000 + 26000)\) | M1 | Uses \(S_n = \frac{n}{2}(a+l)\) with \(a=14000, l=26000\) and \(n=8, 9\) or \(10\). Do not allow ft"s on incorrect \(l\)"s. |
| OR \(S_9 = \frac{n}{2}(2a + (n-1)d) = \frac{9}{2} \times (28000 + 8 \times 1500)\) | Alternatively uses \(S_n = \frac{n}{2}(2a + (n-1)d)\) with \(a=14000, d=1500\) and \(n=8, 9\) or \(10\). Weaker candidates may list the individual salaries. This is acceptable as long as all terms are included. For example \(14000 + 15500 + 17000 + 18500 + 20000 + 21500 + 23000 + 24500 + 26000\) | |
| \(= £180000\) | A1 | Cao (£) 180000. |
| (2 marks) | ||
| (c) Use \(a + (n-1)d\) to find \(A\) | M1 | Use \(l = a + (n-1)d\) to find \(A\). It must be a full method with \(d=1000, l=26000a=A\) and \(n=9, 10\) or \(11\) leading to a value for \(A\) |
| \(A = 17000\) | A1 | \(A=17000\). Accept \(A=17000\) written down for 2 marks as long as no incorrect work seen in its calculation. |
| Use \(S_n = \frac{n}{2}(a+l)\) or \(S_n = \frac{n}{2}(2a+(n-1)d)\) to find \(S\) for Anna. Follow through on their \(A\), but \(l=26000\) and \(n=9, 10\) or \(11\) | M1 | Use \(S_n = \frac{n}{2}(a+l)\) to find \(S\) for Anna. Follow through on their \(A\), but \(l=26000\) and \(n=9, 10\) or \(11\). Alternatively uses \(S_n = \frac{n}{2}(2a + (n-1)d)\) with their numerical value of \(A, d=1000\) and \(n=9, 10\) or \(11\). Accept a series of terms with their value of \(A\), rising in £1000"s up to a maximum of £26000. |
| \(S_{10} = \frac{10}{2}(17000 + 26000)\) OR \(S_{10} = \frac{10}{2}(2 \times 17000 + 9 \times 1000)(= £215000)\) or \(S_{10} = \frac{10}{2}(2 \times 17000 + 9 \times 1000)\) in 10 years | M1A1 | Anna earns \(S_{10} = \frac{10}{2}(17000 + 26000)\) OR \(S_{10} = \frac{10}{2}(2 \times 17000 + 9 \times 1000)\) in 10 years. This is an intermediate answer. There is no requirement to state the value £215 000 |
| Shelim earns \(180000 + 26000\) in 10 years \(= (£206000)\) | B1ft | Shelim earns (b)+26000 in 10 years. This may be scored at the start of part c. |
| Difference = £9000 | A1 | CAO and CSO Difference = £9000 |
| (6 marks) | ||
| (10 marks) |
(a) $14000 + 8 \times 1500 = 14000 + 12000 = £26000$ | M1, A1* | Uses $S = a + (n-1)d$ with $a=14000, d=1500$ and $n=8, 9$ or $10$ in an attempt to find salary in year 9. Accept a sequence written out only if all terms up to year 9 are included—Allow no errors. csa 26000. It is acceptable to write a sequence for both the 2 marks. FYI the terms are $14000, 15500, 17000, 18500, 20000, 21500, 23000, 24500, 26000$
| | **(2 marks)**
(b) $S_n = \frac{n}{2}(a+l) = \frac{9}{2} \times (14000 + 26000)$ | M1 | Uses $S_n = \frac{n}{2}(a+l)$ with $a=14000, l=26000$ and $n=8, 9$ or $10$. Do not allow ft"s on incorrect $l$"s.
OR $S_9 = \frac{n}{2}(2a + (n-1)d) = \frac{9}{2} \times (28000 + 8 \times 1500)$ | | Alternatively uses $S_n = \frac{n}{2}(2a + (n-1)d)$ with $a=14000, d=1500$ and $n=8, 9$ or $10$. Weaker candidates may list the individual salaries. This is acceptable as long as all terms are included. For example $14000 + 15500 + 17000 + 18500 + 20000 + 21500 + 23000 + 24500 + 26000$
$= £180000$ | A1 | Cao (£) 180000.
| | **(2 marks)**
(c) Use $a + (n-1)d$ to find $A$ | M1 | Use $l = a + (n-1)d$ to find $A$. It must be a full method with $d=1000, l=26000a=A$ and $n=9, 10$ or $11$ leading to a value for $A$
$A = 17000$ | A1 | $A=17000$. Accept $A=17000$ written down for 2 marks as long as no incorrect work seen in its calculation.
Use $S_n = \frac{n}{2}(a+l)$ or $S_n = \frac{n}{2}(2a+(n-1)d)$ to find $S$ for Anna. Follow through on their $A$, but $l=26000$ and $n=9, 10$ or $11$ | M1 | Use $S_n = \frac{n}{2}(a+l)$ to find $S$ for Anna. Follow through on their $A$, but $l=26000$ and $n=9, 10$ or $11$. Alternatively uses $S_n = \frac{n}{2}(2a + (n-1)d)$ with their numerical value of $A, d=1000$ and $n=9, 10$ or $11$. Accept a series of terms with their value of $A$, rising in £1000"s up to a maximum of £26000.
$S_{10} = \frac{10}{2}(17000 + 26000)$ OR $S_{10} = \frac{10}{2}(2 \times 17000 + 9 \times 1000)(= £215000)$ or $S_{10} = \frac{10}{2}(2 \times 17000 + 9 \times 1000)$ in 10 years | M1A1 | Anna earns $S_{10} = \frac{10}{2}(17000 + 26000)$ OR $S_{10} = \frac{10}{2}(2 \times 17000 + 9 \times 1000)$ in 10 years. This is an intermediate answer. There is no requirement to state the value £215 000
Shelim earns $180000 + 26000$ in 10 years $= (£206000)$ | B1ft | Shelim earns (b)+26000 in 10 years. This may be scored at the start of part c.
Difference = £9000 | A1 | CAO and CSO Difference = £9000
| | **(6 marks)**
| | **(10 marks)**
---Shelim starts his new job on a salary of £14000. He will receive a rise of £1500 a year for each full year that he works, so that he will have a salary of £15500 in year 2, a salary of £17000 in year 3 and so on. When Shelim's salary reaches £26000, he will receive no more rises. His salary will remain at £26000.
\begin{enumerate}[label=(\alph*)]
\item Show that Shelim will have a salary of £26000 in year 9. [2]
\item Find the total amount that Shelim will earn in his job in the first 9 years. [2]
\end{enumerate}
Anna starts her new job at the same time as Shelim on a salary of £$A$. She receives a rise of £1000 a year for each full year that she works, so that she has a salary of £$(A + 1000)$ in year 2, £$(A + 2000)$ in year 3 and so on. The maximum salary for her job, which is reached in year 10, is also £26000.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the difference in the total amount earned by Shelim and Anna in the first 10 years. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2014 Q7 [10]}}