Word Count : 400

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The given iterative method is: \[ x_{n+1} = \frac{1}{8} \left[ 6x_n + \frac{3N}{x_n} - \frac{x_n^3}{N} \right], \quad n = 0,1,2,\dots \] where \( N \) is a positive constant. We need to determine the value to which this iteration converges and analyze its rate of convergence. --- ### Step 1: Find the Fixed Points To find the limiting value, let \( x_n \to x \) as \( n \to \infty \). Then, at convergence: \[ x = \frac{1}{8} \left[ 6x ___ __________ _______ ___ _________ _______ ______ _______ ________ _________.
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