Ensino Médio ⇒ Proporção Áurea Tópico resolvido

[andrezza]

Sabendo que Φ = 1 + [tex3]\sqrt{5}[/tex3]/2, é correto afirmar que Φ^-1 = Φ - 1.

Resposta

---

Certo

[Planck]

Olá andrezza,

Inicialmente, temos que:

[tex3]\varphi = \frac{1 + \sqrt 5}{2}[/tex3]

Queremos [tex3]\varphi ^{-1}[/tex3], ou seja:

[tex3]\varphi ^{-1} = \frac{1}{ \frac{1 + \sqrt 5}{2}}[/tex3]

[tex3]\varphi ^{-1} = \frac{2}{1 + \sqrt 5} \cdot \frac{(1-\sqrt 5)}{(1- \sqrt5)}[/tex3]

[tex3]\varphi ^{-1} = \frac{2 \cdot(1-\sqrt 5) }{-4}[/tex3]

[tex3]\varphi ^{-1} = \frac{1-\sqrt 5 }{-2}[/tex3]

[tex3]\varphi ^{-1} = \frac{1}{-2} + \frac{-\sqrt5}{-2}[/tex3]

[tex3]\varphi ^{-1} = \frac{\sqrt5}{2}-\frac{1}{2}[/tex3]

[tex3]\varphi ^{-1} +1= \frac{\sqrt5 -1}{2}+1[/tex3]

[tex3]\varphi ^{-1} +1= \frac{\sqrt5 -1}{2}+\frac{2}{2}[/tex3]

[tex3]\varphi ^{-1} +1= \frac{\sqrt5 -1 +2 }{2}[/tex3]

[tex3]\varphi ^{-1} +1= \frac{1 + \sqrt 5 }{2}[/tex3]

[tex3]\varphi ^{-1} = \underbrace{\frac{1 + \sqrt 5 }{2}}_{\varphi} -1[/tex3]

Logo:

[tex3]\boxed{\varphi ^{-1} = {\varphi} -1}[/tex3]