| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: evaluate sum |
| Difficulty | Easy -1.3 This is a straightforward arithmetic sequence question requiring only direct application of the recurrence relation and standard formulas. Part (a) involves simple substitution, part (b) uses the nth term formula, and part (c) applies the sum formula—all routine procedures with no problem-solving or insight required. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04g Sigma notation: for sums of series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(u_2 = -2\), \(u_3 = -7\) and \(u_4 = -12\) | M1, A1 | M1: attempt to use formula correctly at least twice ("subtract 5"); A1: three correct answers |
| Total | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(d = -5\) and arithmetic | B1 | B0 if \(d=-5\) followed by \(3\times(-5)^{99}\); may assume AP if any AP formula seen |
| Uses \(a+(n-1)d\) with \(a=3\) and \(n=100\) to give \(-492\) | M1, A1 | Look for \(3+99\times"d"\) or \(-2+98\times"d"\); A1: \(-492\) (cao) |
| Total | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(S_{100} = \frac{n}{2}(2a+(n-1)d)\) or \(\frac{n}{2}(a+l)\) | M1 | States/uses correct sum formula for AP with \(n=100\), any values for \(a\), \(d\), \(l\) |
| \(S_{100} = \frac{100}{2}(6+99\times-5)\) or \(\frac{100}{2}(3+-492)\) | dM1 | Uses and processes correct formula with \(a=3\) or \(-2\), \(d=\pm5\), ft on their \(l\) |
| \(= -24\,450\) | A1 | Obtains \(-24\,450\) |
| Total | [3] |
# Question 7:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $u_2 = -2$, $u_3 = -7$ and $u_4 = -12$ | M1, A1 | M1: attempt to use formula correctly at least twice ("subtract 5"); A1: three correct answers |
| **Total** | **[2]** | |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $d = -5$ and arithmetic | B1 | B0 if $d=-5$ followed by $3\times(-5)^{99}$; may assume AP if any AP formula seen |
| Uses $a+(n-1)d$ with $a=3$ and $n=100$ to give $-492$ | M1, A1 | Look for $3+99\times"d"$ or $-2+98\times"d"$; A1: $-492$ (cao) |
| **Total** | **[3]** | |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $S_{100} = \frac{n}{2}(2a+(n-1)d)$ or $\frac{n}{2}(a+l)$ | M1 | States/uses correct sum formula for AP with $n=100$, any values for $a$, $d$, $l$ |
| $S_{100} = \frac{100}{2}(6+99\times-5)$ or $\frac{100}{2}(3+-492)$ | dM1 | Uses and processes correct formula with $a=3$ or $-2$, $d=\pm5$, ft on their $l$ |
| $= -24\,450$ | A1 | Obtains $-24\,450$ |
| **Total** | **[3]** | |
---7. A sequence is defined by
$$\begin{aligned}
u _ { 1 } & = 3 \\
u _ { n + 1 } & = u _ { n } - 5 , \quad n \geqslant 1
\end{aligned}$$
Find the values of
\begin{enumerate}[label=(\alph*)]
\item $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$
\item $u _ { 100 }$
\item $\sum _ { i = 1 } ^ { 100 } u _ { i }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2018 Q7 [8]}}