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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