| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2018 |
| Session | October |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Sequence defined by formula |
| Difficulty | Moderate -0.8 This is a straightforward algebraic question requiring substitution into a given formula and basic summation. Part (a) involves setting u₂ = u₄ and solving a simple linear equation for k. Part (b) requires calculating four terms and adding them. No conceptual difficulty or problem-solving insight needed—purely mechanical application of given formulas. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04g Sigma notation: for sums of series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(u_2 = 2k - 3^2\) or \(u_4 = 4k - 3^4\) | M1 | Attempts to use the given formula correctly at least once for \(u_2\) or \(u_4\). Note: \(u_2 = 4k - 3^4\) is M0 |
| \(2k - 9 = 4k - 81 \Rightarrow k = \ldots\) | M1 | Puts their \(u_2 =\) their \(u_4\) and attempts to solve for \(k\) |
| \(k = 36\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(u_1 = 36 - 3^1,\ u_2 = 2(36) - 3^2,\ u_3 = 3(36) - 3^3,\ u_4 = 4(36) - 3^4\) | M1 | Attempts to find values of first 4 terms correctly using their value of \(k\). Allow slips but method and intention should be clear |
| \(\sum_{r=1}^{4} u_r = u_1 + u_2 + u_3 + u_4 = (33 + 63 + 81 + 63)\) | M1 | Adds their first 4 terms. Allow if in terms of \(k\), e.g. \(k-3+2k-3^2+3k-3^3+4k-3^4\ (=10k-120)\) |
| \(\sum_{r=1}^{4} u_r = 240\) | A1 | cao |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $u_2 = 2k - 3^2$ or $u_4 = 4k - 3^4$ | M1 | Attempts to use the given formula correctly at least once for $u_2$ or $u_4$. Note: $u_2 = 4k - 3^4$ is M0 |
| $2k - 9 = 4k - 81 \Rightarrow k = \ldots$ | M1 | Puts their $u_2 =$ their $u_4$ and attempts to solve for $k$ |
| $k = 36$ | A1 | cao |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $u_1 = 36 - 3^1,\ u_2 = 2(36) - 3^2,\ u_3 = 3(36) - 3^3,\ u_4 = 4(36) - 3^4$ | M1 | Attempts to find values of first 4 terms correctly using their value of $k$. Allow slips but method and intention should be clear |
| $\sum_{r=1}^{4} u_r = u_1 + u_2 + u_3 + u_4 = (33 + 63 + 81 + 63)$ | M1 | Adds their first 4 terms. Allow if in terms of $k$, e.g. $k-3+2k-3^2+3k-3^3+4k-3^4\ (=10k-120)$ |
| $\sum_{r=1}^{4} u_r = 240$ | A1 | cao |
---4. A sequence of numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ satisfies
$$u _ { n } = k n - 3 ^ { n }$$
where $k$ is a constant.
Given that $u _ { 2 } = u _ { 4 }$
\begin{enumerate}[label=(\alph*)]
\item find the value of $k$
\item evaluate $\sum _ { r = 1 } ^ { 4 } u _ { r }$
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2018 Q4 [6]}}