| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: find parameter from given term |
| Difficulty | Moderate -0.3 This is a straightforward recurrence relation question requiring simple substitution and algebraic manipulation. Part (a) is direct substitution, part (b) requires one more substitution to verify a given result, and part (c) involves summing four terms and solving a linear equation. While it has multiple parts, each step is routine with no conceptual challenges beyond basic algebra, making it slightly easier than average. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04g Sigma notation: for sums of series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a_2 = 2k-7\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a_3 = 2(2k-7)-7\) or \(4k-14-7 = 4k-21\) | M1, A1cso | Must see \(2(\text{their } a_2)-7\); \(a_2\) must be a function of \(k\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a_4 = 2(4k-21)-7\ (=8k-49)\) | M1 | Attempt to find \(a_4\) using the given rule |
| \(\sum_{r=1}^{4}a_r = k + \text{"\)(2k-7)\("} + \text{"\)(4k-21)\("} + \text{"\)(8k-49)\("}\) | M1 | Attempt the sum of first 4 terms; must use "\(+\)"; use of arithmetic series formulae scores M0M0A0 |
| \(k+(2k-7)+(4k-21)+(8k-49)=15k-77=43 \Rightarrow k=8\) | M1 A1 | Form linear equation in \(k\) using sum and 43; \(k=8\) only (so \(k=\frac{120}{15}\) is A0) |
# Question 7:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a_2 = 2k-7$ | B1 | |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a_3 = 2(2k-7)-7$ or $4k-14-7 = 4k-21$ | M1, A1cso | Must see $2(\text{their } a_2)-7$; $a_2$ must be a function of $k$ |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a_4 = 2(4k-21)-7\ (=8k-49)$ | M1 | Attempt to find $a_4$ using the given rule |
| $\sum_{r=1}^{4}a_r = k + \text{"$(2k-7)$"} + \text{"$(4k-21)$"} + \text{"$(8k-49)$"}$ | M1 | Attempt the sum of first 4 terms; must use "$+$"; use of arithmetic series formulae scores M0M0A0 |
| $k+(2k-7)+(4k-21)+(8k-49)=15k-77=43 \Rightarrow k=8$ | M1 A1 | Form linear equation in $k$ using sum and 43; $k=8$ only (so $k=\frac{120}{15}$ is A0) |
---7. A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by
$$\begin{aligned}
a _ { 1 } & = k \\
a _ { n + 1 } & = 2 a _ { n } - 7 , \quad n \geqslant 1
\end{aligned}$$
where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Write down an expression for $a _ { 2 }$ in terms of $k$.
\item Show that $a _ { 3 } = 4 k - 21$.
Given that $\sum _ { r = 1 } ^ { 4 } a _ { r } = 43$,
\item find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2009 Q7 [7]}}