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
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
2
Explain why the iteration $x_{n+1} = \dfrac{x_n^3 - 5}{2}$ does not converge to the root of $x^3 - 2x - 5 = 0$ near $x = 2.09$ .
Level 3 · 3 marksTry it
3
(a) Show that $x^3 + 4x - 1 = 0$ can be written as $x = \dfrac{1}{x^2 + 4}$ .
(b) Use $x_{n+1} = \dfrac{1}{x_n^2 + 4}$ with $x_0 = 0$ to find the root to 4 decimal places.
Level 3 · 5 marksTry it
4
An iteration $x_{n+1} = g(x_n)$ converges to a root $\alpha$ at which $g'(\alpha) = -0.7$ . State, with a reason, whether the staircase/cobweb diagram is a staircase or a cobweb.
Level 3 · 2 marksTry it
5
The iteration $x_{n+1} = \dfrac12\left(x_n + \dfrac{3}{x_n}\right)$ is used with $x_0 = 2$ .
(a) Find $x_1$ , $x_2$ and $x_3$ , giving each to $4$ decimal places.
(b) State the exact value the sequence converges to.
Level 3 · 4 marksTry it

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