## Exam-Style Questions on Iteration## Problems on Iteration adapted from questions set in previous Mathematics exams. |

## 1. | GCSE Higher |

Using \(x_{n+1}=-5-\frac{6}{x_n^2} \)

with \(x_0 = -1.5 \)

(a) find the values of \(x_1, x_2 \) and \(x_3\)

(b) Explain the relationship between the values of \(x_1, x_2 \) and \(x_3\) and the equation \(x^3+5x^2+6=0 \)

## 2. | GCSE Higher |

(a) A sequence is defined by the following rule where \(u_n\) is the \(n^{th}\) term of the sequence:

$$u_{n+1}=u_n^2-5u_n+21$$If \(u_1=3\) find \(u_2\) and \(u_3\).

(b) A different sequence is defined by the following rule:

$$u_{n+1}=u_n^2-8u_n+17$$If \(u_1=5\) find \(u_2\) and \(u_3\) and \(u_{50}\).

## 3. | GCSE Higher |

Consider the following cubic equation:

$$x^3-7x-5=0$$

An approximate solution can be found by using the following iterative process.

$$x_{n+1}=\frac{(x_n)^3-5}{7}$$(a) Find \(x_2\) and \(x_3\) if \(x_1=-1\)

Work out the solution to 6 decimal places.

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