| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: find specific terms |
| Difficulty | Moderate -0.8 This is a straightforward recurrence relation question requiring only substitution and simplification of surds. Part (a) involves two direct calculations, and part (b) requires two more iterations with basic surd manipulation. No problem-solving insight is needed—just careful arithmetic with nested square roots, making it easier than the average A-level question. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a_2 = (\sqrt{4+3}) = \sqrt{7}\) | B1 | For \(\sqrt{7}\) only |
| \(a_3 = \sqrt{\text{"their 7"}+3} = \sqrt{10}\) | B1ft | Follow through their "7" in correct formula provided they have \(\sqrt{n}\) where \(n\) is an integer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a_4 = \sqrt{10+3} \left(= \sqrt{13}\right)\) | M1 | For attempt to find \(a_4\); must see \(\sqrt{\text{"their"}(a_3)^2 + 3}\). \(a_4 = \sqrt{13}\) provided this follows from their \(a_3\) working |
| \(a_5 = \sqrt{13+3} = 4\) | A1 cso | Must include \(= 4\). Ending at \(\sqrt{16}\) only is A0; ending with \(\pm 4\) is A0. Full list \(2\left(=\sqrt{4}\right), \sqrt{7}, \sqrt{10}, \sqrt{13}, \sqrt{16}=4\) is fine for M1A1 |
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a_2 = (\sqrt{4+3}) = \sqrt{7}$ | B1 | For $\sqrt{7}$ only |
| $a_3 = \sqrt{\text{"their 7"}+3} = \sqrt{10}$ | B1ft | Follow through their "7" in correct formula provided they have $\sqrt{n}$ where $n$ is an integer |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a_4 = \sqrt{10+3} \left(= \sqrt{13}\right)$ | M1 | For attempt to find $a_4$; must see $\sqrt{\text{"their"}(a_3)^2 + 3}$. $a_4 = \sqrt{13}$ provided this follows from their $a_3$ working |
| $a_5 = \sqrt{13+3} = 4$ | A1 cso | Must include $= 4$. Ending at $\sqrt{16}$ only is A0; ending with $\pm 4$ is A0. Full list $2\left(=\sqrt{4}\right), \sqrt{7}, \sqrt{10}, \sqrt{13}, \sqrt{16}=4$ is fine for M1A1 |
---\begin{enumerate}
\item A sequence of positive numbers is defined by
\end{enumerate}
$$\begin{aligned}
a _ { n + 1 } & = \sqrt { } \left( a _ { n } ^ { 2 } + 3 \right) , \quad n \geqslant 1 , \\
a _ { 1 } & = 2
\end{aligned}$$
(a) Find $a _ { 2 }$ and $a _ { 3 }$, leaving your answers in surd form.\\
(b) Show that $a _ { 5 } = 4$\\
\hfill \mbox{\textit{Edexcel C1 2010 Q5 [4]}}