Exam-Style Questions on Iteration

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 $x^3+5x^2+6=0$

Worked Solution

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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

Worked Solution

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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 $u_{50}$.

Worked Solution

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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.

Worked Solution

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