Harrington Wright
01/28/2023 · Senior High School
The sequence of values given by the iterative formula \[ x_{m+1}=\frac{2 x_{m}^{6}+12 x_{n}}{3 x_{m}^{5}+8} \] with initial value \( x_{1}=2 \), converges to \( \alpha \) (i) Use the formula to calculate \( \alpha \) correct to 4 decimal places. Give the result of each iteration to 6 decimal places. (ii) State an equation satisfied by \( \alpha \) and hence find the exact value of \( \alpha \)
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The sequence converges to \( \alpha \approx 1.214 \) (to 4 decimal places). The equation satisfied by \( \alpha \) is \( \alpha^6 + 8 \alpha - 12 = 0 \).
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