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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 large mug of hot tea is cooling on a small lounge table.

The temperature \(n\) minutes after 2pm is \(T_n\) degrees Celsius.

The temperature of the tea (n+1) minutes after 2pm, \(T_{n + 1}\) metres, is given by:

$$T_{n + 1} = A × T_n + 24$$

where \(A\) is a constant.

At 2pm the tea is a piping hot 84°.

At 14:01 temperature of the tea is 80°.

Work out the temperature of the tea at 14:03

Mug of Tea

3.

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 \(u_{50}\).


4.

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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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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