CAIE A-Level Mathematics 3.6.3 Iteration formulae

CAIE A-Level Mathematics 3.6.3 Iteration formulae
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise using and explaining iteration formulae, listing rounded iterates and justifying roots to a specified accuracy.

How this is tested

  • 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

Question 7(d)

[Maximum number: 3]

The polynomial p(x) is defined by

p(x)=2x4+kx3+kx2+17x+18,\mathrm{p}(x)=2 x^{4}+k x^{3}+k x^{2}+17 x+18,

where k is a constant. It is given that (x+2) is a factor of p(x).

Use an iterative formula, based on the equation in part (b), to find the value of β\beta correct to 3 significant figures. Give the result of each iteration to 5 significant figures.