| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: find parameter from given term |
| Difficulty | Moderate -0.5 This is a straightforward recurrence relation question requiring simple substitution to find a₂ = 4k - 3, then a₃ = 4(4k-3) - 3, followed by solving a linear equation a₁ + a₂ + a₃ = 66. It's slightly easier than average as it involves only basic algebraic manipulation with no conceptual challenges beyond understanding the recurrence notation. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04g Sigma notation: for sums of series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((a_2 =)\ 4k-3\) | B1 | \(4k-3\) cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a_3 = 4(4k-3)-3\) | M1 | An attempt to find \(a_3\) from iterative formula \(a_3=4a_2-3\). Condone bracketing errors |
| \(\sum_{r=1}^{3}a_r = k+4k-3+4(4k-3)-3 = ..k\pm...\) | M1 | Attempt to sum \(a_1, a_2\) and \(a_3\) to get a linear expression in \(k\). (Sum of Arithmetic series is M0) |
| \(21k-18=66 \Rightarrow k=...\) | dM1 | Sets their linear expression to 66 and solves for \(k\). Dependent on previous M mark |
| \(k=4\) | A1 | cao \(k=4\) |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(a_2 =)\ 4k-3$ | B1 | $4k-3$ cao |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a_3 = 4(4k-3)-3$ | M1 | An attempt to find $a_3$ from iterative formula $a_3=4a_2-3$. Condone bracketing errors |
| $\sum_{r=1}^{3}a_r = k+4k-3+4(4k-3)-3 = ..k\pm...$ | M1 | Attempt to sum $a_1, a_2$ and $a_3$ to get a linear expression in $k$. (Sum of Arithmetic series is M0) |
| $21k-18=66 \Rightarrow k=...$ | dM1 | Sets their linear expression to 66 and solves for $k$. Dependent on previous M mark |
| $k=4$ | A1 | cao $k=4$ |
---A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by
$$\begin{array} { l l }
a _ { n + 1 } = 4 a _ { n } - 3 , & n \geqslant 1 \\
a _ { 1 } = k , & \text { where } k \text { is a positive integer. }
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Write down an expression for $a _ { 2 }$ in terms of $k$.
Given that $\sum _ { r = 1 } ^ { 3 } a _ { r } = 66$
\item find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2014 Q3 [5]}}