CAIE A-Level Mathematics 3.6.3 Iteration formulae
Practise using and explaining iteration formulae, listing rounded iterates and justifying roots to a specified accuracy.
- Syllabus
- 2028–2030
- Course
- Mathematics 9709
- Level
- A2
Exam points
- use x_n+1=F(x_n) repeatedly, showing iterates to the requested number of decimal places
- rearrange the fixed-point equation to show a convergent iteration solves the original equation
- justify the final rounded root by enough iterations or a sign change in the rounding interval
3.6.3—Iteration formulae question 1
[Maximum number: 4]
Question (a)
(a)
Show that if a sequence of values given by the iterative formula
xn+1=21cos−1(−e−xn)
converges, then it converges to the root of the equation in part (a).
[ 1 ]
State the equation x=1/2 ^-1(-e^-x) and rearrange to obtain 2 x=-e^x
Or rearrange 2 x=-e^x as x=1/2 ^-1(-e^-x) and state x_n+1=1/2 ^-1(-e^-x_n)
B1
Question (b)
(b)
Use the iterative formula given in part (c) to calculate x correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
[ 3 ]
Use the iterative process correctly at least once
M1
Obtain final answer 0.978
A1
Show sufficient iterations to at least 5 d. p. to justify 0.978 to 3 d.p. or show there
is a sign change in the interval (0.9775,0.9785)
A1
E.g.
1, 0.97376, 0.97903,0.97796, 0.97813
0.95, 0.98395, 0.97697, 0.97838, 0.97809
0.9, 0.99475, 0.97480, 0.97882, 0.97800, 0.97817
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