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Given: $x_n = (x_{n - 1})^2 + x_{n - 1}$ for $x \geq 2,$ and $x_1 = \frac{1}{2}.$

 

Problem: Prove that $(\frac{1- x_1}{x_1})(\frac{1- x_2}{x_2}) \cdots (\frac{1- x_{n-1}}{x_{n-1}}) < 2$ for all $x \geq 1.$

 Jun 20, 2021
edited by MathProblemSolver101  Jun 20, 2021
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Did you mean ... for \(n\ge2\) rather than ...for \(x\ge2 \)?

 

If so:

 Jun 20, 2021
edited by Alan  Jun 20, 2021

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