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Alpha is the root of the equation E(x) = x^3 - 5x -50 = 0

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October 7, 2010
algebra unsolvedalgebra

Problem Statement

If α\alpha is the real root of the equation E(x)=x3−5x−50=0E(x) = x^3 - 5x -50 = 0 such that xn+1=(5xn+50)1/3x_{n+1} = (5x_n + 50)^{1/3} and x1=5x_1 = 5, where nn is a positive integer, prove that:
(a) xn+13−α3=5(xn−α)x_{n+1}^3 - \alpha^3 = 5(x_n - \alpha)
(b) α<xn+1<xn.\alpha < x_{n+1} < x_n.
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