Bonjour ;
1)
[tex]x = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + ...} } } } \\\\\
x^2 = 1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + ...} } } } \\\\
x + 1 = 1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + ...} } } }[/tex]
[tex]x^2 = x + 1 \Rightarrow x^2 - x - 1 = 0 .[/tex]
2)
[tex]x^2 - x - 1 = x^2 - 2 * \dfrac{1}{2} * x + ( \dfrac{1}{2})^2 - ( \dfrac{1}{2})^2 - 1 \\\\ =(x^2 - 2 * \dfrac{1}{2} * x + ( \dfrac{1}{2})^2) - \dfrac{1}{4} - 1 = (x - \dfrac{1}{2})^2 - \dfrac{5}{4} .[/tex]
3)
[tex]x^2 - x - 1 = 0 \Rightarrow \Delta = 1 + 4 = 5 \\\\ \Rightarrow x_1 = \dfrac{1 - \sqrt{5} }{2} \ \textless \ 0 \\\\ x_2 = \dfrac{1 + \sqrt{5} }{2} \ \textgreater \ 0 ; [/tex]
donc :
[tex]x = \dfrac{1 + \sqrt{5} }{2} .[/tex]
