# 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.

The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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