C1 June 2010 Q5
5. A sequence of positive numbers is defined by\[\begin{aligned} a_{n+1} &= \sqrt{\left(a_n^2 + 3\right)}, \qquad n \geqslant 1, \\ a_1 &= 2 \end{aligned}\]
| Scheme | Marks |
|---|---|
| \(a_2 = \left(\sqrt{4 + 3}\right) = \sqrt{7}\) | B1 |
| \(a_3 = \sqrt{\text{"their } 7\text{"} + 3} = \sqrt{10}\) | B1ft |
| (2) |
Notes
1st B1: for \(\sqrt{7}\) only
2nd B1ft: follow through their “7” in correct formula provided they have \(\sqrt{n}\), where \(n\) is an integer.
ALT Formula: Some may state (or use) \(a_n = \sqrt{3n + 1}\) leading to \(a_5 = \sqrt{3\times 5 + 1} = 4\).
This will get marks in (a) [if correct values are seen] and can score the M1 in (b) if \(a_n = \sqrt{3n + 1}\) or \(a_4 = \sqrt{13}\) are seen.
\(\pm\sqrt{\ }\) If \(\pm\sqrt{\ }\) appear any where ignore in part (a) and withhold the final A mark only
| Scheme | Marks |
|---|---|
| \(a_4 = \sqrt{10 + 3}\ \left(= \sqrt{13}\right)\) | M1 |
| \(a_5 = \sqrt{13 + 3} = 4 \quad\) * | A1 cso |
| (2) | |
| (4 marks) |
Notes
M1: for an attempt to find \(a_4\). Should see \(\sqrt{\text{"their"}\left(a_3\right)^2 + 3}\). Must see evidence for M1.
\(a_4 = \sqrt{13}\) provided this follows from their \(a_3\) working or answer is sufficient
A1cso: for a correct solution (M1 explicit) must include the \(= 4\).
Ending at \(\sqrt{16}\) only is A0 and ending with \(\pm 4\) is A0.
Ignore any incorrect statements that are not used e.g. common difference \(= \sqrt{3}\)
Listing: A full list: \(2\ \left(= \sqrt{4}\right),\ \sqrt{7},\ \sqrt{10},\ \sqrt{13},\ \sqrt{16} = 4\) is fine for M1A1
ALT Formula: Some may state (or use) \(a_n = \sqrt{3n + 1}\) leading to \(a_5 = \sqrt{3\times 5 + 1} = 4\).
This will get marks in (a) [if correct values are seen] and can score the M1 in (b) if \(a_n = \sqrt{3n + 1}\) or \(a_4 = \sqrt{13}\) are seen.
\(\pm\sqrt{\ }\) If \(\pm\sqrt{\ }\) appear any where ignore in part (a) and withhold the final A mark only