| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2009 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Non-linear or complex iterative formula convergence |
| Difficulty | Standard +0.3 This is a straightforward iterative formula question requiring mechanical calculation of iterations until convergence (standard calculator work) followed by algebraic manipulation to find the exact value by setting x_{n+1} = x_n = α. Both parts are routine A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams1.09d Newton-Raphson method |
| Answer | Marks |
|---|---|
| Use the iterative formula correctly at least once | M1 |
| State final answer 2.78 | A1 |
| Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in an appropriate function in (2.775, 2.785) | A1 |
| [3] |
| Answer | Marks |
|---|---|
| State a suitable equation, e.g. \(x = \frac{3}{4}x + \frac{15}{x^3}\) | B1 |
| State that the exact value of \(a\) is \(\sqrt[4]{60}\), or equivalent | B1 |
| [2] |
**(i)**
| Use the iterative formula correctly at least once | M1 |
| State final answer 2.78 | A1 |
| Show sufficient iterations to at least 4 d.p. to justify its accuracy to 2 d.p., or show there is a sign change in an appropriate function in (2.775, 2.785) | A1 |
| [3] |
**(ii)**
| State a suitable equation, e.g. $x = \frac{3}{4}x + \frac{15}{x^3}$ | B1 |
| State that the exact value of $a$ is $\sqrt[4]{60}$, or equivalent | B1 |
| [2] |3 The sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 3 x _ { n } } { 4 } + \frac { 15 } { x _ { n } ^ { 3 } }$$
with initial value $x _ { 1 } = 3$, converges to $\alpha$.\\
(i) Use this iterative formula to find $\alpha$ correct to 2 decimal places, giving the result of each iteration to 4 decimal places.\\
(ii) State an equation satisfied by $\alpha$ and hence find the exact value of $\alpha$.
\hfill \mbox{\textit{CAIE P3 2009 Q3 [5]}}