A-level · Numerical methods

Iterative methods (x = g(x))Exam-style questions

Solving f(x) = 0 by rearranging it into x = g(x) and iterating x_(n+1) = g(x_n) from a starting value — why some rearrangements converge and others diverge, and reading staircase and cobweb diagrams.

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20 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
(a) Show that $x^3 - 7x + 2 = 0$ can be written as $x = \dfrac{x^3 + 2}{7}$ .
(b) Using $x_{n+1} = \dfrac{x_n^3 + 2}{7}$ with $x_0 = 0$ , find the root near $0$ to 3 decimal places.
Level 2 · 5 marksTry it
2
The iteration $x_{n+1} = \sqrt[3]{2x_n + 5}$ is used with $x_0 = 2$ to find a root of $x^3 - 2x - 5 = 0$ . Find $x_1$ , $x_2$ and $x_3$ , giving each to 4 decimal places.
Level 2 · 3 marksTry it
3
Show that the equation $x^{2} - 5x + 3 = 0$ can be rearranged into the form

$x = \dfrac{x^{2} + 3}{5}$ (2 marks)
Level 2 · 2 marksTry it
4
The iteration $x_{n+1} = \sqrt{5x_{n} - 2}$ is used with $x_{0} = 2$ .

Find $x_{1}$ , $x_{2}$ and $x_{3}$ , giving each to 4 decimal places. (3 marks)
Level 2 · 3 marksTry it
5
The iteration $x_{n+1} = \sqrt{3x_n + 4}$ converges to a limit $\alpha$ . Find, with integer coefficients, the equation that $\alpha$ satisfies, and hence state the value of $\alpha$ .
Level 3 · 4 marksTry it

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