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Question Number 117095 by bemath last updated on 09/Oct/20

	Answered by Olaf last updated on 09/Oct/20

	$\frac{u_{n + 1}}{2} + 1 = u_{n}$ $\frac{l_{1}}{2} + 1 = l_{1} \Rightarrow l_{1} = 2$ $v_{n} = \frac{1}{v_{n + 1}} + 2$ $l_{2} = \frac{1}{l_{2}} + 2$ $l_{2}^{2} - 2 l_{2} - 1 = 0$ $l_{2} = \frac{2 \pm \sqrt{8}}{2} \Rightarrow l_{2} = 1 + \sqrt{2}$ $Left side \to \frac{l_{1}}{l_{2}} = \frac{2}{1 + \sqrt{2}} = 2 (\sqrt{2} - 1)$ $w_{n} = 1 - \frac{w_{n + 1}}{2}$ $l_{3} = 1 - \frac{l_{3}}{2} \Rightarrow l_{3} = \frac{2}{3}$ $z_{n} = 2 - \frac{1}{z_{n + 1}}$ $l_{4} = 2 - \frac{1}{l_{4}}$ $l_{4}^{2} - 2 l_{4} + 1 = 0$ $l_{4} = 1$ $Right side \to \frac{l_{3}}{l_{4}} = \frac{2 / 3}{1} = \frac{2}{3}$ $Left side + Right side = 2 \sqrt{2} - 2 + \frac{2}{3}$ $\Rightarrow \frac{6 \sqrt{2} - 4}{3} (A)$
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