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Exam-Style Questions on IterationProblems 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
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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