Question

( fi ) al cuadrado es fi+1?

Answer 1

Respuesta:

$\boxed{\bold{ {\Phi}^{2} =\Phi + 1}} \: \: \to \: \: \bold{verdadero}$

Explicación paso a paso:

Comprobar que:

$\boxed{ \bold{{\Phi }^{2} = \Phi + 1}}$

Si se sabe que:

$\bold{\Phi = \frac{1 + \sqrt{5} }{2}}$

Elevamos al cuadrado:

$\bold{{\Phi}^{2} = \left ( \frac{1 + \sqrt{5} }{2} \right) ^{2}}$

$\bold{{\Phi}^{2} = \frac{(1 + \sqrt{5} )^{2} }{ {(2)}^{2} } }$

$\bold{{\Phi}^{2} = \frac{ {1}^{2} + 2(1)( \sqrt{5} ) + {( \sqrt{5}) }^{2} }{4 } }$

$\bold{{\Phi}^{2} = \frac{1 + 2 \sqrt{5} + 5}{4 }}$

$\bold{{\Phi}^{2} = \frac{6 + 2 \sqrt{5} }{4 } }$

$\bold{{\Phi}^{2} = \frac{2(3+ \sqrt{5} )}{2(2) } }$

$\bold{{\Phi}^{2} = \frac{3+ \sqrt{5}}{2} }$

$\bold{{\Phi}^{2} = \frac{2 +1+ \sqrt{5}}{2 }}$

$\bold{{\Phi}^{2} = \frac{2}{2} + \frac{1+ \sqrt{5}}{2 }}$

$\bold{ {\Phi}^{2} = 1+ \red{ \frac{1+ \sqrt{5}}{2 }}}$

$\bold{Recordar \: \: que: } \: \boxed{ \bold{ \Phi = \frac{1 + \sqrt{5} }{2}}}$

$\bold{{\Phi}^{2} = 1+ \red{ \Phi }}$

$\boxed{\bold{ {\Phi}^{2} =\Phi + 1}}$