The sequence of values given by the iterative formula $$\(x_{n+1}=\frac{6+8 x_{n}}{8+x_{n}^{2}}\)$$ with initial value $$\(x_{1}=2\)$$ converges to $$\(\alpha\)$$. State an equation satisfied by $$\(\alpha\)$$ and hence determine the exact value of $$\(\alpha\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics
IGCSE&ALevel
CAIE
Exam No:9709_w20_qp_23 Year:2020 Question No:5(b)

Answer:

State equation \(x=\frac{6+8 x}{8+x^{2}}\) or equivalent using \(\alpha\)
Obtain \(\sqrt[3]{6}\) or exact equivalent

Knowledge points:

2.6.2 understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation

Solution:

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